  In this paper we restrict ourselves to the particular case where the
pseudogroup is $\Gamma \ltimes G$ given by the action of a dense subgroup
$\Gamma$ 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 $\Gamma \ltimes G$. The study of a generalization
of this particular case shall be the main purpose of this paper.
