The main theorem in Chapter 11 of "Topology and Groupoids" is that if the group $G$ acts properly and discontinuously on a Hausdorff space $X$ which admits a universal cover, then the fundamental groupoid of the orbit space $X/G$ is isomorphic to the orbit groupoid $\pi_1(X)/\!/G$. (Any improvement suggested? The key point is that if a group acts on a space $X$ then it also acts on the fundamental groupoid $\pi_1 (X)$, so that the orbit groupoid is defined by an obvious universal property.) The proof goes by verifying the appropriate universal property. The problem of calculating this orbit groupoid is also dealt with there. As an example, the fundamental group of the symmetric square of a space is calculated. See also arXiv:math/0212271 for an account of some of this Chapter.
A basic paper on orbit groupoids is
J. Taylor, "Quotients of groupoids by the action of a group", Math. Proc. Camb. Phil. Soc., 103, (1988) 239-249.
Later: I should also mention the earlier work, but not using the language of groupoids, of M.A. Armstrong, in two papers:
"Lifting homotopies through fixed points", Proc. Roy. Soc. Edinburgh A93 (1982) 123--128.
Also II 96 (1984) 201--205.
Ross Geoghegan in his review in MathSciNet of the second paper in 1986 writes: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years."