# What is Out(G-mod) for a finite group G?

Following the notation of Etingof-Nikshych-Ostrik what is Out(G-mod) for a finite group G?

That is what are all bimodule cateogries over the fusion category G-mod of complex G-modules which have the property that they're just G-mod as a left (resp. right) G-module up to equivalence of bimodules? I think this is the same as the group of tensor autoequivalences of G-mod which do not come from conjugating by 1-dimensional objects, but ENO only prove that in the case where there are no 1-dimensional objects.

If you have an automorphism of the group G then you get an automorphism of G-mod. I'm not totally sure though if outer automorphisms of G necessarily give outer automorphisms (for example could you have an outer automorphism that fixed all conjugacy classes?) and I also don't know whether all outer automorphisms come up this way.

The reason that I'm asking is that I want to understand the relationship between Out(C) and Out(D) where C and D are Morita equivalent fusion categories. However, I realized I don't have enough examples where I understand what Out(C) actually is.

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Nontrivial outer automorphisms can fix all conjugacy classes. See the 1947 paper by G.E. Wall (ams.org/mathscinet-getitem?mr=25461) for examples. Wall says that such examples were first given by Burnside in 1912, but that paper is apparently too old to appear on MathSciNet. – Evan Jenkins Jul 15 '10 at 23:24

The group $\operatorname{Out}(G\operatorname{-Mod})$ is equivalent to the group $\operatorname{Aut}_{\textbf{Tw}}(k[G])$ of gauge equivalence classes of twisted automorphisms of the Hopf algebra $k[G]$ defined by Davydov in Twisted automorphisms of Hopf algebras. In Davydov's other paper, Twisted automorphisms of group algebras, he describes this group in the case that $\lvert G \rvert$ is relatively prime to 6. (The proposed description I gave in the earlier version of this answer fails in general because $\operatorname{Out}(G)$ is not in general a normal subgroup of $\operatorname{Aut}_{\textbf{Tw}}(k[G])$.)
First, we note that since $G\operatorname{-Mod}$ is symmetric, any inner tensor autoequivalence is automatically tensor equivalent to the identity, so it suffices to describe the group of isomorphism classes of tensor autoequivalences. Let $\omega: G\operatorname{-Mod} \to \operatorname{Vect}$ be the usual fiber functor. Any tensor autoequivalence $F: G\operatorname{-Mod} \stackrel{\cong}{\to} G\operatorname{-Mod}$ gives a new (tensor) fiber functor $\omega \circ F: G\operatorname{-Mod} \to \operatorname{Vect}$. This gives $G\operatorname{-Mod}$ the structure of the category of representations of some Hopf algebra; it is shown in the paper On families of triangular Hopf algebras by Etingof and Gelaki that this Hopf algebra will be a twist of the original one, and thus $F$ induces a twisted automorphism of $H$. Conversely, any twisted automorphism of $F$ gives rise to an autoequivalence of $G\operatorname{-Mod}$. Natural transformations of monoidal functors correspond to gauge equivalences of twisted automorphisms. Thus, we can identify isomorphism classes of monoidal autoequivalences of $G-\operatorname{Mod}$ with gauge equivalence classes of twisted automorphisms of $k[G]$.