If a group acts on a groupoid , then there is an orbit groupoid which is a groupoid with trivial -action and with a -morphism universal for -morphisms to groupoids with trivial -action. So in principle is obtained from by identifying and for all , .
Note that if a group acts on a space then it has an induced action on the fundamental groupoid , and so there is an induced morphism
\alpha: \Pi_1 X /\! / G \to \Pi_1 (X/G).
So there is interest in when this morphism is an isomorphism.
is an isomorphism if is Hausdorff, has a universal cover, and the action of on is discontinuous.
R. Brown, Topology and groupoids, Booksurge, 2006, Chapter 11.
J. Taylor, “Quotients of groupoids by the action of a group, Math. Proc. Camb. Phil. Soc., 103 (1988) 239–249.