A Vershik-Kerov theorem for wreath products

Let $G_{n,k}$ be the group of permutations of $\{1,2,\ldots, kn\}$ that permutes the first $k$ symbols arbitrarily, then the next $k$ symbols and so on through the last $k$ symbols. Finally the $n$ blocks of size $k$ are permuted in an arbitrary way. For $\sigma$ chosen uniformly in $G_{n,k}$, let $L_{n,k}$ be the length of the longest increasing subsequence in $\sigma$. For $k,n$ growing, we determine that the limiting mean of $L_{n,k}$ is asymptotic to $4\sqrt{nk}$. This is different from parallel variations of the Vershik-Kerov theorem for colored permutations.