In the title, $R$ stands for the **hyperfinite III_{1} factor.**

An order two element $\alpha\in Out(M)$ ($M$ any factor) has an invariant $c(\alpha)\in H^3(\mathbb Z/2,S^1)=\mathbb Z/2$.

Q:Is $c$ the only invariant of $\alpha$ (up to conjugation) when $M$ is the hyperfiniteIII_{1}factor?

*Background:*

The group $Out(M)$ is π_{0}(G) for G the following 2-group: the monoidal category of invertible $M$-$M$-bimodules, and isometries. That 2-group has π_{1}(G)=S^{1}
and, as any 2-group, it is classified by a characteristic class in H^{3}(π_{0}(G),π_{1}(G)).
The characteristic class $c(\alpha)\in H^3(\mathbb Z/2,S^1)$ is pulled back from that universal characteristic class $c_{\text{univ.}}\in H^3(Out(M),S^1)$ along the homomorphism $\mathbb Z/2\to Out(M)$.

*Generalization:*

Let $G$ be any finite group (or maybe compact group... or maybe infinite ameanable...). It is true that conjugacy classes of injective homomorphisms from $G\to Out(R)$ are classified by $H^3(G,S^1)$?