Proof. Let $b_D$ be a Brauer correspondent of B in $D\mathrm {C}_G(D)$ . Then by [Reference Murray35, Lemma 2.2], we may choose E in such a way that $b_D^{E\mathrm {C}_G(D)}$ is real with defect pair $(D,E)$ . In other words, $(D,b_D,E)$ is a Sylow B-subtriple in the notation of [Reference Murray35]. Since B is not nilpotent, there exists a so-called essential subgroup $Q\le D$ in the fusion system $\mathcal {F}$ of B. Then Q is a Klein four-group and there exists a unique B-subpair $(Q,b_Q)\le (D,b_D)$ (see [Reference Sambale37, Theorem 1]). Moreover, $b_Q$ is nilpotent with defect group $\mathrm {C}_D(Q)=Q$ (see [Reference Aschbacher, Kessar and Oliver1, Theorem IV.3.19]). Since Q is essential,

$$ \begin{align*}\mathrm{N}_G(Q,b_D)/\mathrm{C}_G(Q)\cong \operatorname{Aut}_{\mathcal{F}}(Q)\cong S_3.\end{align*} $$

The block $B_Q:=b_Q^{\mathrm {N}_G(Q,b_Q)}$ has defect group $\mathrm {N}_D(Q)\cong D_8$ by [Reference Aschbacher, Kessar and Oliver1, Theorem IV.3.19]. By [Reference Navarro36, Corollary 9.21], $B_Q$ is the only block of $\mathrm {N}_G(Q,b_Q)$ that covers $b_Q$ . Since every subgroup of $S_3$ has trivial Schur multiplier, each $\psi \in \operatorname {Irr}(b_Q)$ extends to its inertial group. The number of extensions is determined by Gallagher’s theorem. If $\psi $ is $\mathrm {N}_G(Q,b_Q)$ -invariant, $\operatorname {Irr}(B_Q)$ contains three extensions of $\psi $ . Since $k(B_Q)=5$ , there can be at most one such character $\psi $ . If $\mathrm {N}_G(Q,b_Q)$ has two orbits of length $2$ in $\operatorname {Irr}(b_Q)$ , then we would get six characters in $\operatorname {Irr}(B_Q)$ . Therefore, the four characters in $\operatorname {Irr}(b_Q)$ distribute into orbits of length $1$ and $3$ under the action of $\mathrm {N}_G(Q,b_Q)$ . In particular, $\operatorname {Irr}(b_Q)$ contains (at least) three characters with the same F-S indicator.

The possible extended defect groups E were determined in [Reference Murray35, Proposition 4.1]. Suppose that our claim is false. Then we are in case (b), that is, $E\cong D*C_4$ is a central product. By [Reference Murray35, Lemma 2.6], $b_Q$ is real and has extended defect group $\mathrm {C}_E(Q)= Q*E\cong C_4\times C_2$ . However now, [Reference Murray34, Theorem 1.7] implies that exactly two characters in $\operatorname {Irr}(b_Q)$ have F-S indicator $1$ . This contradicts the observation above.

Proof. Note that $B_0(H)$ is isomorphic to a (nonreal) block $B_0(H)\otimes b$ of $H\times C_3$ , where $b\cong F$ is a nonprincipal block of $C_3$ . Clearly, $B_0(H)$ and $B_0(H)\otimes b$ have the same fusion system. Let

$$ \begin{align*}B:=(B_0(H)\otimes b)^G\end{align*} $$

be the Fong–Reynolds correspondent of $B_0(H)\otimes b$ in G. By [Reference Linckelmann32, Theorem 6.8.3], B is Puig equivalent to $B_0(H)\otimes b$ (that is, the blocks have the same source algebra). This implies that B is Morita equivalent to $B_0(H)$ and both blocks have the same fusion system (see [Reference Linckelmann32, Theorem 8.7.1]). In particular, B has defect group D.

Let $\operatorname {Irr}(b)=\{\theta \}$ . Then $\chi :=(1_H\times \theta )^G\in \operatorname {Irr}(B)$ is a real character and therefore B is real. Moreover, B is not the principal block of G since otherwise, B cannot cover $B_0(H)\otimes b$ . Therefore, an extended defect group of B must be a Sylow $2$ -subgroup of G, because $|G:H\times C_3|=2$ . Let $e\in E\setminus D$ and $x\in S_3$ an involution. Then, $D\langle ex\rangle $ is a Sylow $2$ -subgroup of G and

$$ \begin{align*}E\to D\langle ex\rangle,\quad g\mapsto\begin{cases} g&\text{if }g\in D,\\ gx&\text{if }g\notin D \end{cases} \end{align*} $$

is an isomorphism.

Proof of Theorem 1.1

By Proposition 2.1 and [Reference Murray35, Table 2], it remains to construct examples for each Morita equivalence class and each defect pair. By the remark above, we may assume that B is not nilpotent. Then by [Reference Macgregor33, Theorem 2.1], B is Morita equivalent to $B_0(H)$ , where H is one of the following groups:

1. (1) $\operatorname {PGL}(2,q)$ with $|q-1|_2=2^{d-1}$ ;

2. (2) $\operatorname {PGL}(2,q)$ with $|q+1|_2=2^{d-1}$ ;

3. (3) $\operatorname {PSL}(2,q)$ with $|q-1|_2=2^d$ ;

4. (4) $\operatorname {PSL}(2,q)$ with $|q+1|_2=2^d$ ;

5. (5) $A_7$ with $d=3$ .

Note that for every $d\ge 3$ , there exists a prime q congruent to $\pm 1+2^d$ modulo $2^{d+1}$ by Dirichlet’s theorem. Thus, appropriate groups H exist for every d. Moreover, if $q\equiv 1+2^d\pmod {2^{d+1}}$ , then $q^2\equiv 1+2^{d+1}\pmod {2^{d+2}}$ . This will be used later on.

For the cases (2) and (5), Murray [Reference Murray35, Table 2] has shown that only $E\cong D\times C_2$ is possible. Here, we take $G=H\times S_3$ as explained above. In case (4), we find $E\cong D_{2^{d+1}}$ in [Reference Murray35, Table 2]. Since

$$ \begin{align*} H\cong\operatorname{SL}(2,q)\mathrm{Z}(\operatorname{GL}(2,q))/\mathrm{Z}(\operatorname{GL}(2,q))\le\operatorname{PGL}(2,q), \end{align*} $$

we can take $\hat {H}:=\operatorname {PGL}(2,q)$ with the required properties.

Suppose now that case (1) occurs. By the remark above, we find a prime p such that $q=p^2\equiv 1+2^d\pmod {2^{d+1}}$ . Let $\sigma $ be the Frobenius automorphism $\mathbb {F}_q\to \mathbb {F}_q$ , $x\mapsto x^p$ . Then the semilinear group $\hat {H}:=H\rtimes \langle \sigma \rangle $ has Sylow $2$ -subgroup $E\cong C_{2^{d-1}}\rtimes C_2^2$ , where $C_2^2$ acts faithfully on $C_{2^{d-1}}$ . (This is type (e) in [Reference Murray35, Proposition 4.1].)

Next, consider case (3). The choice $\hat {H}:=\operatorname {PGL}(2,q)$ realises $E\cong D_{2^{d+1}}$ . We may therefore assume that $q=p^2$ as above. Then there exists a subgroup $\hat {H}$ of ${\operatorname {PGL}(2,q)\rtimes \langle \sigma \rangle }$ of index $2$ with semidihedral Sylow $2$ -subgroup $E\cong SD_{2^{d+1}}$ . (This group is denoted by $\operatorname {PGL}(2,q)^*$ in [Reference Gorenstein19].) Finally, let $d\ge 4$ . Here, $\hat {H}:=H\rtimes \langle \sigma \rangle $ has Sylow $2$ -subgroup $E\cong C_{2^{d-1}}\rtimes C_2^2$ as above.

Proof. Let $\Omega :=\{x\in G:x^2=1\}$ . Note that the equation in Conjecture 3.2 is also true for $x=1$ since B is not principal. Recall that $\Phi $ vanishes on the elements of even order. By the definition of F-S indicators and Conjecture 3.2,

$$ \begin{align*} \epsilon(\Phi)&=\frac{1}{|G|}\sum_{g\in G}\Phi(g^2)=\frac{1}{|G|}\sum_{x\in\Omega}\sum_{h\in\mathrm{C}_G(x)^0}\Phi(h^2)=\sum_{x\in\Omega/G}[\Phi_{\mathrm{C}_G(x)},1_{\mathrm{C}_G(x)}]\\ &=\sum_{x\in\Omega/G}|x^G\cap E\setminus D|=|\Omega\cap E\setminus D|=|\{x\in E\setminus D:x^2=1\}|.\\[-50pt] \end{align*} $$

Proof. By Brauer’s formula [Reference Brauer8, Theorem 4A] (see also [Reference Sambale41, Lemma 2]),

$$ \begin{align*} \epsilon(\Phi_\varphi^x)=\sum_{\psi\in\operatorname{Irr}(b)}\epsilon(\psi)d^x_{\psi\varphi}. \end{align*} $$

Since $\Omega _x=\{y\in \mathrm {C}_G(x):y^2=x\}$ , we may assume that $x\in \mathrm {Z}(G)$ and $B=b$ . By Brauer’s second main theorem, the claim only depends on $\varphi $ , and not on B. For $g\in G^0$ , we compute

$$ \begin{align*} \sum_{\varphi\in\operatorname{IBr}(G)}\epsilon(\Phi_\varphi^x)\varphi(g)&=\sum_{\chi\in\operatorname{Irr}(G)}\epsilon(\chi)\sum_{\varphi\in\operatorname{IBr}(G)}d_{\chi\varphi}^x\varphi(g)=\sum_{\chi\in\operatorname{Irr}(G)}\epsilon(\chi)\chi(xg)\\ &=|\{y\in G:y^2=xg\}|=|\{y\in\mathrm{C}_G(g):y^2=xg\}| \end{align*} $$

[Reference Isaacs26, page 49]. Since g has odd order, there exists a unique power $\sqrt {g}$ of g such that $\sqrt {g}^2=g$ . It is easy to check that the map $\{y\in \mathrm {C}_G(g):y^2=x\}\to \{y\in \mathrm {C}_G(g):y^2=xg\}$ , $y\mapsto y\sqrt {g}$ is a bijection. Hence,

$$ \begin{align*}\sum_{\varphi\in\operatorname{IBr}(G)}\epsilon(\Phi_\varphi^x)\varphi(g)=|\{y\in\mathrm{C}_G(g):y^2=x\}|=|\mathrm{C}_G(g)\cap\Omega_x|=\pi(g)\end{align*} $$

for all $g\in G^0$ . Therefore, $\epsilon (\Phi _\varphi ^x)$ is the multiplicity of $\varphi $ in $\pi $ .

Now assume that there is no $e\in E\setminus D$ such that $e^2=x$ . Then [Reference Murray35, Lemma 1.3] implies

$$ \begin{align*}0=\sum_{\chi\in\operatorname{Irr}(B)}\epsilon(\chi)\chi(x)=\sum_{\chi\in\operatorname{Irr}(B)}\epsilon(\chi)\sum_{\varphi\in\operatorname{IBr}(b)}d_{\chi\varphi}^x\varphi(1)=\sum_{\varphi\in\operatorname{IBr}(b)}\epsilon(\Phi_\varphi^x)\varphi(1),\end{align*} $$

and the second claim follows.

Proof. As in the proof of Lemma 4.1, we may apply Brauer’s formula. Since b has defect pair $(\mathrm {C}_D(x),\mathrm {C}_E(x))$ and

$$ \begin{align*}\{e\in E\setminus D:e^2=x\}=\{e\in\mathrm{C}_E(x)\setminus\mathrm{C}_D(x):e^2=x\},\end{align*} $$

we may assume that $x\in \mathrm {Z}(G)$ and $B=b$ . Now D is abelian and Conjecture 3.1 holds for B by [Reference Sambale41, Theorem 10]. Since every Brauer correspondent $\beta $ of B in a section of G is nilpotent with abelian defect groups, Conjecture 3.1 also holds for $\beta $ . The claim follows from [Reference Sambale41, Theorem 13].

Proof. Let $(D,E)$ be a defect pair of B. It is well known that B has exactly four irreducible characters of height $0$ (see [Reference Sambale38, Theorem 8.1]). By [Reference Gow24, Theorem 5.1], at least one such character has F-S indicator $1$ . Since nonreal characters come in pairs of the same degree, B has two or four real irreducible characters of height $0$ .

To prove the second claim, it suffices to show that $D/D'$ has a complement in $E/D'$ by [Reference Gow24, Theorem 5.6]. Since $D/D'\cong C_2^2$ , we may assume that $E/D'\cong C_4\times C_2$ by way of contradiction. In particular, $|E:E'|\ge 8$ . A theorem of Alperin–Feit–Thompson asserts that the number of involutions in E is congruent to $3$ modulo $4$ (see [Reference Isaacs26, Theorem 4.9]). By the remark after [Reference Isaacs26, Theorem 4.9], the number of involutions in D is congruent to $1$ modulo $4$ . Hence, there exists an involution $x\in E\setminus D$ . However, then $\langle x\rangle D'/D'$ is a complement of $D/D'$ in $E/D'$ , which gives a contradiction.

Proof. Let $\epsilon _1,\ldots ,\epsilon _4$ be the F-S indicators of the height $0$ characters $\lambda _1,\ldots ,\lambda _4\in \operatorname {Irr}(B)$ . We may choose a B-subsection $(x,b)$ such that $|\langle x\rangle |=4$ and b has defect pair $(\langle x\rangle ,\mathrm {C}_E(x))$ . Clearly, b is nilpotent with abelian defect group $\langle x\rangle $ . Let $\operatorname {IBr}(b)=\{\varphi _x\}$ . By the orthogonality relations of generalised decomposition numbers (see [Reference Sambale38, Theorem 1.14]), we have $d_{\lambda _i,\varphi _x}^x=\pm 1$ for $1\le i\le 4$ and $d_{\chi ,\varphi _x}^x=0$ for $\chi \in \operatorname {Irr}(B)\setminus \{\lambda _1,\ldots ,\lambda _4\}$ . These numbers depend on the ordinary decomposition matrix of B.

Case 1: $l(B)=1$ .

Here, B is nilpotent with decomposition matrix $(1,1,1,1,2)^{\mathrm {t}}$ . We may choose our labelling in such a way that $(d_{\lambda _1,\varphi _x}^x,\ldots ,d_{\lambda _4,\varphi _x}^x)=(1,1,-1,-1)$ . Similarly, there are elements $y,xy\in D$ of order $4$ such that $(d_{\lambda _1,\varphi _y}^y,\ldots ,d_{\lambda _4,\varphi _y}^y)=(1,-1,1,-1)$ and $(d_{\lambda _1,\varphi _{xy}}^{xy},\ldots ,d_{\lambda _4,\varphi _{xy}}^{xy})=(1,-1,-1,1)$ with appropriate labelling. If there exists no ${e\in E\setminus D}$ such that $e^2\in \{x,y,xy\}$ , then $(\epsilon _1,\ldots ,\epsilon _4)=(1,1,1,1)$ by Propositions 4.2 and 5.1. Now suppose that $e^2=x$ for some $e\in E\setminus D$ . Then e has order $8$ and it follows easily that $E\cong \{Q_{16},SD_{16}\}$ . In both cases, we have $|\{e\in E\setminus D:e^2=x\}|=2$ and $(\epsilon _1,\ldots ,\epsilon _4)=(1,1,0,0)$ by Propositions 4.2 and 5.1.

Now suppose that $l(B)>1$ . By Macgregor [Reference Macgregor33, Corollary 2.4], there are two cases to consider.

Case 2: B is Morita equivalent to the principal block of $\operatorname {SL}(2,3)$ .

Here, $l(B)=3$ and B has decomposition matrix

$$ \begin{align*} Q:=\begin{pmatrix} 1&.&.\\ .&1&.\\ .&.&1\\ 1&1&1\\ 1&1&.\\ 1&.&1\\ .&1&1 \end{pmatrix}. \end{align*} $$

We see that $\lambda _4$ is real and $\epsilon _4=1$ . The orthogonality relations imply that $(d_{\lambda _1,\varphi _x}^x,\ldots ,d_{\lambda _4,\varphi _x}^x)=\pm (1,1,1,-1)$ . If there exists $e\in E\setminus D$ with $e^2=x$ (that is, $E\in \{Q_{16}, SD_{16}\}$ ), then $(\epsilon _1,\ldots ,\epsilon _4)=(1,1,1,1)$ and otherwise $(\epsilon _1,\ldots ,\epsilon _4)=(0,0,1,1)$ by Propositions 4.2 and 5.1 (after relabelling if necessary).

Case 3: B is Morita equivalent to the principal block of $\operatorname {SL}(2,5)$ .

Here, B has decomposition matrix

$$ \begin{align*} Q:=\begin{pmatrix} 1&.&.\\ 1&1&.\\ 1&.&1\\ 1&1&1\\ .&1&.\\ .&.&1\\ 2&1&1 \end{pmatrix}. \end{align*} $$

It follows that $\lambda _1$ , $\lambda _4$ are real and $(d_{\lambda _1,\varphi _x}^x,\ldots ,d_{\lambda _4,\varphi _x}^x)=\pm (1,-1,-1,1)$ . If ${E\in \{Q_{16},SD_{16}\}}$ , then $(\epsilon _1,\ldots ,\epsilon _4)=(1,0,0,1)$ , and $(\epsilon _1,\ldots ,\epsilon _4)=(1,1,1,1)$ otherwise.

Proof. We reuse the notation from the proof of Proposition 5.2.

Case 1: $l(B)=1$ .

By Proposition 5.2, $(\epsilon _1,\ldots ,\epsilon _4)=(1,1,0,0)$ . Let $\operatorname {Irr}(B)=\{\lambda _1,\ldots ,\lambda _4,\psi \}$ and $\operatorname {IBr}(B)=\{\varphi \}$ . Then,

$$ \begin{align*}2+2\mu=\epsilon_1d_{\lambda_1,\varphi}+\cdots+\epsilon_4d_{\lambda_4,\varphi}+\mu d_{\psi,\varphi}\ge 0\end{align*} $$

with equality if and only if $E\cong Q_{16}$ by [Reference Murray35, Lemma 1.3]. Hence, $\mu =-1$ if ${E\cong Q_{16}}$ and $\mu =1$ if $E\cong \textit {SD}_{16}$ . Examples for both cases can be constructed by the remark after Proposition 2.2. The groups are $\mathtt {SmallGroup}(48,18)$ for $E\cong Q_{16}$ and $\mathtt {SmallGroup}(48,17)$ for $E\cong \textit {SD}_{16}$ in the small groups library [42].

Now let $l(B)=3$ and $\operatorname {IBr}(B)=\{\varphi _1,\varphi _2,\varphi _3\}$ . Let $\psi _1,\psi _2,\psi _3\in \operatorname {Irr}(B)$ be the characters of height $1$ . Let $\mu _i:=\epsilon (\psi _i)$ for $i=1,2,3$ .

Case 2: B is Morita equivalent to the principal block of $\operatorname {SL}(2,3)$ .

By Proposition 5.2, $(\epsilon _1,\ldots ,\epsilon _4)=(1,1,1,1)$ . Assume first that $E\cong Q_{16}$ . Then Lemma 4.1 implies

$$ \begin{align*}d_{\lambda_1,\varphi_i}+\cdots+d_{\lambda_4,\varphi_i}+\mu_1 d_{\psi_1,\varphi_i}+\mu_2 d_{\psi_2,\varphi_i}+\mu_3 d_{\psi_3,\varphi_i}\ge 0\end{align*} $$

for $i=1,2,3$ . The shape of the decomposition matrix of B yields $\mu _1=\mu _2=\mu _3=-1$ as claimed. For the purpose of constructing an infinite family of examples, let q be an odd prime and $H:=\operatorname {SL}(2,q)$ . Let $\zeta \in \mathbb {F}_{q^2}^\times $ of order $2(q-1)$ . Then,

$$ \begin{align*} \hat{H}:= H\bigg\langle \begin{pmatrix}\zeta&0\\ 0 &\zeta^{-1}\end{pmatrix} \biggr\rangle\le \operatorname{SL}(2,q^{2}) \end{align*} $$

is a nonsplit extension with Sylow $2$ -subgroup $E\cong Q_{2^{d+1}}$ . Thus, we can apply Proposition 2.2 to the pair $(H,\hat {H})$ . For $q=3$ , we end up with the (unique) nonprincipal block of $G=\mathtt {SmallGroup}(144,124)$ .

Now assume that $E\cong \textit {SD}_{16}$ . Here, we need to investigate the generalised decomposition matrix $Q^z$ with respect to a B-subsection $(z,b_z)$ , where $\mathrm {Z}(D)=\langle z\rangle $ . The columns of $Q^z$ lie in the orthogonal complement of the $\mathbb {Z}$ -module spanned by the columns of the ordinary decomposition matrix and the column $d_{.,\varphi _x}^x$ . It is easy to find a basis of this $\mathbb {Z}$ -module. Therefore, there exists an integral matrix $S\in \operatorname {GL}(3,\mathbb {Q})$ such that

$$ \begin{align*}Q^z=\begin{pmatrix} 1&.&.\\ .&1&.\\ .&.&1\\ 1&1&1\\ -1&-1&.\\ -1&.&-1\\ .&-1&-1 \end{pmatrix}S.\end{align*} $$

By Lemma 4.1, $(\epsilon _1,\ldots ,\epsilon _4,\mu _1,\mu _2,\mu _3)Q^z=(0,0,0)$ . After we multiply both sides with $S^{-1}$ , we get $\mu _1=\mu _2=\mu _3=1$ as desired. To construct examples, let $H:=\operatorname {SL}(2,q)$ for an odd prime q. Let $\mathbb {F}_q^\times =\langle \zeta \rangle $ . The conjugation with $\bigl (\begin {smallmatrix}0&\zeta \\1&\zeta \end {smallmatrix}\bigr )\in \operatorname {GL}(2,q)$ induces an automorphism $\alpha $ on H of order $2$ . Then $\hat {H}:=H\rtimes \langle \alpha \rangle $ has Sylow $2$ -subgroup $E\cong SD_{2^{d+1}}$ , so we can apply Proposition 2.2. For $q=3$ , this gives the nonprincipal block of $G=\mathtt {SmallGroup}(144,125)$ .

Case 3: B is Morita equivalent to the principal block of $\operatorname {SL}(2,5)$ .

Here we have $(\epsilon _1,\ldots ,\epsilon _4)=(1,0,0,1)$ with the labelling of the proof of Proposition 5.2. The decomposition matrix shows further that $\varphi _2=\overline {\varphi _3}$ and therefore $\mu _1=\mu _2=0$ . As before, we have

$$ \begin{align*} (1+\mu_3)(2\varphi_1(1)+\varphi_2(1)+\varphi_3(1))=\lambda_1(1)+\lambda_4(1)+\mu_3\psi_3(1)\ge 0 \end{align*} $$

with equality if and only if $E\cong Q_{16}$ . Hence, $\mu _3=-1$ if $E\cong Q_{16}$ and $\mu _3=1$ if $E\cong \textit {SD}_{16}$ . The former case occurs for a nonprincipal block of $G=\mathtt {SmallGroup}(720,414)$ and the latter for a nonprincipal block of $G=\mathtt {SmallGroup}(720,415)$ (same construction as in Case 2 with $q=5$ ).

Proof. Let $l(B)=1$ . Then B is nilpotent and the generalised decomposition matrix of B coincides with the character table of D. This shows that $k(B)=16$ and exactly four characters are $2$ -rational. The possible groups E can be computed with GAP and examples can be found among the groups of order $96$ as in Proposition 2.2. If B is the principal block (that is, $E=D$ ), then $\operatorname {Irr}(B)=\operatorname {Irr}(G/\mathrm {O}_{2'}(G))=\operatorname {Irr}(D)$ . In this case, the claim is easy to check. Otherwise, the F-S indicators are determined by [Reference Sambale41, Theorem 10]. We note that the embedding of D in E is not always unique, but the F-S indicators are independent of this embedding (in our situation).

Now let $l(B)=3$ . Suppose first that B is the principal block. By a result of Brauer [Reference Brauer7, Theorem 1], we have $\operatorname {Irr}(B)=\operatorname {Irr}(G/\mathrm {O}_{2'}(G))=\operatorname {Irr}(D\rtimes C_3)$ . The F-S indicators can be computed easily here. Now let $E\ne D$ . We argue as in Proposition 2.1 to exclude most candidates for E. Let $(D,b_D)$ be a fixed Brauer pair. By [Reference Sambale41, Proposition 8(i)], the extended stabiliser has the form $\mathrm {N}_G(D,b_D)^*=E\mathrm {N}_G(D,b_D)$ . It can be checked that $\mathrm {N}_G(D,b_D)/\mathrm {C}_G(D)$ is isomorphic to a Sylow $3$ -subgroup S of $\operatorname {Aut}(D)$ . Moreover, the normaliser of S in $\operatorname {Aut}(S)$ has three conjugacy classes of involutions. Hence, there are at most four possible actions of E on D (including the trivial action). This excludes the cases $\mathrm {id}\in \{4,12,24,25,26,31,32\}$ . Now let $\mathrm {id}=3$ , that is, $E\cong C_8\times C_4$ . Then $b_D$ is real and nilpotent. By the first part of the proof, $b_D$ has exactly $12$ nonreal characters. Under the action of $\mathrm {N}_G(D,b_D)$ , the $16$ characters in $\operatorname {Irr}(b_D)$ distribute into five orbits of length $3$ and one orbit of length $1$ . In particular, the number of nonreal characters in $\operatorname {Irr}(b_D)$ cannot be $12$ , which gives a contradiction.

Next assume that $\mathrm {id}=35$ . Here, E acts on D by inverting its elements. In particular, E centralises $D_1:=\langle x^2:x\in D\rangle \cong C_2^2$ . Fix a B-subpair $(D_1,b_1)$ . Then $b_1$ has defect pair $(\mathrm {C}_D(D_1),\mathrm {C}_E(D_1))=(D,E)$ . In particular, $b_1$ is real. Since S does not centralise $D_1$ , $b_1$ is nilpotent. By the first part of the proof, $b_1$ has exactly eight characters with F-S indicator $-1$ . However, this leads to a contradiction by considering the action of $\mathrm {N}_G(D,b_D)$ in $\operatorname {Irr}(b_1)$ as above. This leaves the cases $\mathrm {id}\in \{11,21,33,34\}$ . In all of those, E splits over D. Hence, all F-S indicators are nonnegative by [Reference Gow24, Theorem 5.6].

By [Reference Eaton, Kessar, Külshammer and Sambale13], the decomposition matrix of B is

$$ \begin{align*}Q:=\begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ 1&1&1\\ 1&1&1\\ 1&1&1\\ 1&1&1\\ 1&1&1 \end{pmatrix}.\end{align*} $$

Up to conjugation, there are five nontrivial B-subsections $(x,b_x)$ , $(y,b_y)$ , $(y^{-1},b_y)$ , $(z,b_z)$ and $(z^{-1},b_z)$ , where x is an involution and $y,z$ have order $4$ . Thus, the complex conjugation fixes exactly four columns of the generalised decomposition matrix $\hat {Q}$ . By Brauer’s permutation lemma, there are exactly four $2$ -rational characters. We stress that the $2$ -conjugate characters can still be real since y might be conjugate to $y^{-1}$ via an element not fixing $b_y$ . By the shape of Q, we may assume that the first four characters are $2$ -rational. By [Reference Gow24, Theorem 5.3], two or four of them have F-S indicator $1$ .

By [Reference Sambale39, Theorem 15], B is isotypic to the principal block of $H:=D\rtimes C_3$ . This implies via [Reference Sambale40, Proposition 7.3] that $\hat {Q}$ coincides up to basic sets with the generalised decomposition matrix of H. However, since $l(b_x)=l(b_y)=l(b_z)=1$ , the columns of $\hat {Q}$ corresponding to $x,y,z$ are uniquely determined up to signs. Using the orthogonality relations, the signs can be chosen such that

$$ \begin{align*}\hat{Q}=\begin{pmatrix} 1&0&0&\epsilon_x&\epsilon_y&\epsilon_y&\epsilon_z&\epsilon_z\\ 0&1&0&\epsilon_x&\epsilon_y&\epsilon_y&\epsilon_z&\epsilon_z\\ 0&0&1&\epsilon_x&\epsilon_y&\epsilon_y&\epsilon_z&\epsilon_z\\ 1&1&1&3\epsilon_x&-\epsilon_y&-\epsilon_y&-\epsilon_z&-\epsilon_z\\ 1&1&1&-\epsilon_x&(-1+2i)\epsilon_y&(-1-2i)\epsilon_y&\epsilon_z&\epsilon_z\\ 1&1&1&-\epsilon_x&(-1-2i)\epsilon_y&(-1+2i)\epsilon_z&\epsilon_z&\epsilon_z\\ 1&1&1&-\epsilon_x&\epsilon_y&\epsilon_y&(-1+2i)\epsilon_z&(-1-2i)\epsilon_z\\ 1&1&1&-\epsilon_x&\epsilon_y&\epsilon_y&(-1-2i)\epsilon_z&(-1+2i)\epsilon_z \end{pmatrix},\end{align*} $$

where $\epsilon _x,\epsilon _y,\epsilon _z\in \{\pm 1\}$ and $i=\sqrt {-1}$ . Since $b_x$ , $b_y$ and $b_z$ are nilpotent with abelian defect group, we can apply Proposition 4.2 for each E. This gives a linear system on the vector of F-S indicators. We have checked by computer that there is a unique solution (up to permuting the first three characters). Examples are given by $G=\mathtt {SmallGroup}(288,a)$ , where $a=67,407,406,405$ for $\mathrm {id}=11,21,33,34$ , respectively.