The fundamental group of the orbit space

Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. Let G¯$\bar G$ be the closure of G in Homeo(X). The class of an orbit O of G is the union of all orbits having the same closure as O. We denote by X/G˜$X/\widetildeG$ the space of classes of orbits called the orbit class space. In this paper, we study the fundamental group of the spaces X/G, X/G¯$X/\bar G$ and X/G˜$X/\widetildeG$