  The topological Molino's description of equicontinuous foliated spaces,
studied by the first author and Moreira Galicia, gives conditions to reduce
their study to the particular case where the holonomy pseudogroup can be
represented by a pseudogroup on some local group $G$ generated by some of its
local left translations (a $G$-foliated space). That description is sharpened
in this paper by introducing a foliated action of a compact topological group
on the resulting $G$-foliated space, like in the case of Riemannian foliations.
Moreover a $C^\infty$ version is also studied. The triviality of this compact
group characterizes compact minimal $G$-foliated spaces, which are also
characterized by their foliated homogeneity in the $C^\infty$ case. We also
give an example where the projection of the Molino's description is not a
principal bundle, and another example of positive topological codimension where
the foliated homogeneity cannot be checked by only comparing pairs of
leaves---in the case of zero topological codimension, weak solenoids with this
property were given by Fokkink and Oversteegen, and later by Dyer, Hurder and
Lukina.
