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.