Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. Let
$\bar G$
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
$X/\widetildeG$
the space of classes of orbits called the orbit class space. In this paper, we study the fundamental group of the spaces X/G,
$X/\bar G$
and
$X/\widetildeG$