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](http://arxiv.org/PS_cache/math/pdf/0111/0111139v1.pdf) 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](http://arxiv.org/abs/math/0007196).


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.



Are there two groups which are categorically Morita equivalent but only one of which is simple
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](http://arxiv.org/abs/0810.0032), 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).


