  Let $\Gamma$ be a finite group. Consider the wreath product $G_n := \Gamma^n
\rtimes S_n$ and the subgroup $K_n := \Delta_n \times S_n\subseteq G_n$, where
$S_n$ is the symmetric group and $\Delta_n$ is the diagonal subgroup of
$\Gamma^n$. For certain values of $n$ (which depend on the group $\Gamma$), the
pair $(G_n, K_n)$ is a Gelfand pair. It is not known for all finite groups
which values of $n$ result in Gelfand pairs. Building off the work of
Benson--Ratcliff, we obtain a result which simplifies the computation of
multiplicities of irreducible representations in certain tensor product
representations, then apply this result to show that for $\Gamma = S_k, \ k
\geq 5$, $(G_n,K_n)$ is a Gelfand pair exactly when $n = 1,2$.
