Group extensions and actions on categories Let G and H be two groups. There is a one-to-one correspondence between:
(i) an (isomorphism class of) extension of G by H, i.e. an exact sequence of group morphisms $1\to H\to E\to G\to 1$;
(ii) an (isomorphism class of) action of the group G on the category of H-sets.
Actually, this can even be made an equivalence of categories.
This is probably well-known by category-theoretists, but I found no reference on the subject. Does anybody know of an article or book where it has been treated ?
Lots of thanks, in advance.
 A: The beginning of SGA 7, Exposé VII, essentially proves an equivalence between extensions of $G$ by $H$ and monoidal functors from $G$ (thought of as a discrete monoidal category) to the monoidal category of $H$-bitorsors. The correspondence here takes an extension $1 \to H \to E \stackrel{\pi}{\to} G \to 1$ to the functor sending $g \in G$ to the fiber $\pi^{-1}(g)$, which naturally carries the structure of an $H$-bitorsor via the multiplication in $E$.
This latter category can be identified with the monoidal category of autoequivalences of the category of $H$-sets: an autoequivalence of the category of $H$-sets is given as tensoring by an $H$-bitorsor, and natural isomorphisms between autoequivalences correspond to isomorphisms of $H$-bitorsors. Off the top of my head, I don't know of a reference that describes this latter equivalence explicitly.
A: Such extensions are what D. Conduché called crossed 2-modules (in French: modules croisés de longueur 2) in his 1983 paper: Modules croisés généralisés de longueurs 2, J. Pure and Appl. Alg. Vol. 34, Issues 2-3 (1984), pp. 155-178. My guess is that an action of $G$ on the category of $H$-sets amounts to what the same author called "non-abelian pre-crossed complex" (in French: complexe pré-croisé non abélien) in the same paper. The equivalence you talked about is then Théorème 2.6 of the above mentioned reference.
A: Another relevant paper is 
R. Brown, G. Danesh-Naruie, J.P.L. Hardy  ``The fundamental groupoid as a
topological groupoid'', Proc. Edinburgh Math. Soc. 19 (1975)
237-244.
Section 4 explains that an extension $1 \to B \to E \to G \to 1$ is isomorphic to an exact sequence determined by a fibration of groupoids $G \ltimes A \to G$ where $A$ is a groupoid which is a $G$-module. In the case in point, $A$ is the action groupoid determined by the action of  $E$ on the right of $G$. 
Understanding of this really requires the notion of covering morphism of groupoids and the equivalence of categories between covering morphisms of $G$ and actions of $G$ on sets. I feel this notion of covering morphism is somewhat neglected. However in the book 
Higgins, P.J. Notes on categories and groupoids, Mathematical Studies, Volume 32. van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195.
the notion is used to prove a generalisation of Grushko's theorem. And fibrations of groupoids are also useful algebraically, and topologically! 
