Let $\mathbb{G}$ be a 2-group, by which I mean a strict monoidal category in which all objects are invertible (up to coherent isomorphisms) and all morphisms are invertible (strictly).
- What is the definition of $\mathbb{G}$-torsor, thought of as a groupoid equipped with a $\mathbb{G}$-action?
- Let $G$ be a group, let $\mathbb{B} G$ be $G$ regarded as a one-object groupoid, and let $\mathbb{A}$ be the automorphism 2-group of $\mathbb{B} G$ (i.e. the 2-group whose objects are the group automorphisms of $G$ and whose morphisms $f_0 \to f_1$ are those elements $g$ in $G$ such that $g f_0 (x) = f_1 (x) g$ for all $x$ in $G$). There is an obvious action of $\mathbb{A}$ on $\mathbb{B} G$. Does it make $\mathbb{B} G$ into an $\mathbb{A}$-torsor?
- If the answer to (2) is no, then what is the connection with the old notion of "gerbe with band"? My impression is that $\mathbb{B} G$ is supposed to be a gerbe with band $\mathrm{OutAut}(G)$ (over the point).