Are there two groups which are categorically Morita equivalent but only one of which is simple Can you find two finite groups G and H such that their representation categories are Morita equivalent (which is to say that there's an invertible bimodule category over these two monoidal categories) but where G is simple and H is not.  The standard reference for module categories and related notions is this paper of Ostrik's
This is a much stronger condition than saying that C[G] and C[H] are Morita equivalent as rings (where C[A_7] and C[Z/9Z] gives an example, since they both have 9 matrix factors).  It is weaker than asking whether a simple group can be isocategorical (i.e. have representation categories which are equivalent as tensor categories) with a non-simple group, which was shown to be impossible by Etingof and Gelaki.
Matt Emerton asked me this question when I was trying to explain to him why I was unhappy with any notion of "simple" for fusion categories.  It's of interest to the study of fusion categories where the dual even Haagerup fusion category appears to be "simple" while the principal even Haagerup fusion category appears to be "not simple" yet the two are categorically Morita equivalent.
 A: 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).
A: The farthest I got thinking about this problem (and I haven't thought about it all that much) is that module categories over $\mathbb C[G]$ are classified in Section 3.4 of Ostrik - Module categories, weak Hopf algebras, and modular invariants.  They correspond to pairs $K$ a subgroup of $G$ and a choice of central extension of $K$ (or equivalently, a certain cohomology class).  In the case where there's no central extension, the dual category is some sort of Hecke algebra category $\text{$\mathbb C[K\backslash G/K]$-mod}$ that I've never totally understood.  Also I don't know how to modify that construction when you introduce the central extension.  Anyway, modulo understanding those issues, the question comes down to when a twisted Hecke algebra category $\text{$\mathbb C[K\backslash G/K]$-mod}$ can be equivalent as a tensor category to $\text{$\mathbb C[H]$-mod}$ for some group $H$.
A: 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).
A: 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.
