Let
$A\,\in \,{{M}_{n}}\left( \mathbb{R} \right)$
be an invertible matrix. Consider the semi-direct product
${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$
where the action of
$\mathbb{Z}$
on
${{\mathbb{R}}^{n}}$
is induced by the left multiplication by
$A$
. Let
$\left( \alpha ,\,\tau \right)$
be a strongly continuous action of
${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$
on a
${{C}^{*}}$
-algebra
$B$
where
$\alpha$
is a strongly continuous action of
${{\mathbb{R}}^{n}}$
and
$\tau$
is an automorphism. The map
$\tau$
induces a map
$\widetilde{\tau }\,\text{on}\,\text{B}\,{{\rtimes }_{\alpha }}\,{{\mathbb{R}}^{n}}$
. We show that, at the
$K$
-theory level,
$\tau$
commutes with the Connes–Thom map if
$\det \left( A \right)\,>\,0$
and anticommutes if
$\det \left( A \right)\,>\,0$
. As an application, we recompute the
$K$
-groups of the Cuntz–Li algebra associated with an integer dilation matrix.