Suppose that $N$ is a normal $p$-subgroupof a finite group $G$ and let$G^0$ be the set of elements of $G$ whose$p$-part lies in $N$. We prove the existenceof a canonical basis ${\rm IBr}(G, N)$of the space of complex class functions of $G$defined on $G^0$, such that the restriction $\chi^0$of any irreducible complex character $\chi$ of $G$is a linear combination$\sum_{\phi\in{\rm IBr}(G, N)} d_{\chi \phi} \phi$ ofthe elements of this basis, where the $d_{\chi \phi}$are non-negative integers.Furthermore, if we write$\Phi_\phi=\sum_{\chi} d_{\chi \phi}\chi$,then the $\Phi_\phi$ form the K\"ulshammer--Robinson${\Bbb Z}$-basis ofthe ${\Bbb Z}$-module generated by the characters afforded by the $N$-projective $RG$-modules,where $R$ is a certain complete discrete valuationring. By using these `decomposition numbers',it is possible to define a linking in the set of theirreducible complex characters of $G$. 1991 Mathematics Subject Classification: 20C15, 20C20.