Quotients of singular foliations and Lie 2-group actions

Androulidakis–Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of assignment: if a foliated manifold $(M,\mathcal{F}\_M)$ is the quotient $(P,\mathcal{F}\_P)$ along surjective submersion with connected fibers, then same true for corresponding groupoids. For quotients by Lie group action, analogue statement holds under suitable assumptions, yielding 2-group action on