On locally trivial extensions of topological spaces by a pseudogroup

In this paper we restrict ourselves to the particular case where the pseudogroup is $\mathrm{\Gamma }⋉G$ given by the action of a dense subgroup $\mathrm{\Gamma }$ of a Lie group $G$ acting on $G$ by left translations. For a Riemannian foliation $F$ on a complete Riemannian manifold $M$ which is transversally parallelizable in the sense of Molino, let $X$ be the space of leaves closures. The holonomy pseudogroup of $F$ is an example of a locally trivial extension of $X$ by $\mathrm{\Gamma }⋉G$. The study of a generalization of this particular case shall be the main purpose of this paper.