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The following wonderful paper:

Manfred Droste, Michèle Giraudet, Rüdiger Göbel, All Groups are Outer Automorphism Groups of Simple Groups Journal of the London Mathematical Society 64 (2001) 565–575, doi:10.1112/S0024610701002484

shows that every group $G$ is isomorphic to $\mathrm{Out}(H)$ for some simple group $H$. The natural question arises about how unique this prescription is. What can be said about two simple groups (usually infinite in my case) with the same outer automorphism group? Anything along the lines of Morita/monoidal Morita or similar equivalence?

I am interested in a general answer, of course. But partial answers and comments are also welcome. Please, note that I am not an algebraist, so even if the answer involves advanced notions, I would appreciate a description of those as simple as possible. Thank you.

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    $\begingroup$ A result of Ashot Minasyan goes some way to giving a negative answer to your question, at least for countable groups: Theorem. Let $C$ be an arbitrary countable group. Then for every non-elementary torsion-free word hyperbolic group $H$ there exists a torsion-free group $N$ satisfying the following properties: 1. $N$ is a $2$-generated quotient of $H$; 2. $N$ has two conjugacy classes; 3. $\operatorname{Out}(N)\cong C$. See Theorem 1.5 of Minasyan, A. "Groups with finitely many conjugacy classes and their automorphisms." Comm. Math. Hel. (2009) MR2495795 arxiv.org/abs/0704.0091 $\endgroup$
    – user1729
    Mar 29, 2017 at 9:41
  • $\begingroup$ Thanks for the comment. It remains to show that there exists no reasonable cohomological or similar equivalence relation containing all such $N$. I agree that the contrary would be a very strong and unexpected result. $\endgroup$
    – Bedovlat
    Mar 29, 2017 at 9:49

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