TITLE: Are there two groups which are categorically Morita equivalent but only one of which is simple
QUESTION [26 upvotes]: 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.

REPLY [12 votes]: 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.