Are there two groups which are categorically Morita equivalent but only one of which is simple

Categorically Morita equivalent groups were studied by Deepak Naidu in Categorical Morita equivalence for group-theoretical categories. He obtained there a complete description of Morita equivalent groups. It is also shown that simple groups are categorically Morita rigid.

I think an answer to your question is given in Naidu, Nikshych, and Witherspoon - Fusion subcategories of representation categories of twisted quantum doubles of finite groups, theorem 1.1.

Subcategories of the double $D(G)$ are given by pairs of normal subgroups $K$, $N$ in $G$ which centralize each other, together with the datum of a bicharacter $K\times N \to \mathbb C^\times$.

So in particular if $G$ has no normal subgroups and $H$ does, then you're going to find that $D(G)$ has no nontrivial subcategories, while $D(H)$ will (one can take $K$ the normal subgroup in $H$, $N=\{id\}$, and the bicharacter $K\to \mathbb C^\times$ to be trivial, I guess).

Wouldn't it follow that the quantum doubles of the two groups are isomorphic? Would this help to set the question? (Sorry for posting this as an answer, didn't manage to leave it as a comment).