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!