Homomorphisms from a finite group into wreath products

Let $G$ be a finite group, $A$ a finite abelian group. Each homomorphism $\varphi :G\to A\wr S_n$ induces a homomorphism $\bar{\varphi }:G\to A$ in a natural way. We show that as $\varphi$ is chosen randomly, then the distribution of $\bar{\varphi }$ is close to uniform. As application we prove a conjecture of T. M"uller on the number of homomorphisms from a finite group into Weyl groups of type $D_n$.