  It is well known that the pair $(\mathcal{S}_n,\mathcal{S}_{n-1})$ is a
Gelfand pair where $\mathcal{S}_n$ is the symmetric group on $n$ elements. In
this paper, we prove that if $G$ is a finite group then $(G\wr \mathcal{S}_n,
G\wr \mathcal{S}_{n-1}),$ where $G\wr \mathcal{S}_n$ is the wreath product of
$G$ by $\mathcal{S}_n,$ is a Gelfand pair if and only if $G$ is abelian.
