## Groups

Let $G$ be a finite group. An **involutive automorphism** on $G$ is an automorphism $i\colon G \to G$ such that $i^2 = 1_G$.

**Question 1.** What classifies involutive automorphisms on a given (non-abelian) finite group? In particular, is there any classification in terms of group cohomology?

Note that the involutive automorphisms needn't form a group, just like the involutions (the "inner involutive automorphisms", if you wish) don't necessarily.

Observe the following: Whenever $h = i(h)^{-1}$, then $j(g) = h^{-1}i(g)h$ is again involutive.

**Question 1.1.** Is the set of involutive automorphisms a torsor for some group?

Given two involutive automorphisms $i$ and $j$, then $ji$ is an automorphism of $G$, but I don't see any guarantee that it is inner, let alone conjugation by an element $h$ satisfying $i(h) = h^{-1}$. Now famously the number of involutions is given by a formula involving irreducible characters and Frobenius-Schur indicators, but I'm not aware of any such result for involutive automorphisms.

## Fusion rings

Let $R$ be a unital ring with a finite basis $B$ containing the neutral element such that all structure constants are nonnegative. Such rings are called **based rings**. Morphisms of based rings are required to map basis elements onto basis elements.

A **fusion ring** is a ring structure on $\mathbb{Z}^n$ with basis $\{(1,0,\dots), (0,1,0,\dots), \dots\}$, together with an anti-involution $i$ on $R$ such that for every basis element $b$, the multiplicity of the neutral element in $b \cdot i(b)$ is exactly $1$. (The Grothendieck ring of a fusion category is always a fusion ring, hence the name. The extra condition on the chosen anti-involution encodes the existence of duals in the category.)

It is easy to check that every finite group gives rise to a fusion ring, where the chosen anti-involution is inversion. But there are also many fusion rings that don't come from groups, for example the Fibonacci ring with basis $\{1, \tau\}$ satisfying $\tau \cdot \tau = 1 + \tau$.

**Questions 2-2.1.** What classifies involutive automorphisms on fusion rings? (And the same further questions as for groups.)