  Every singular foliation has an associated topological groupoid, called
holonomy groupoid (see arXiv:math/0612370). In this note we exhibit some
functorial properties of this assignment: if a foliated manifold
$(M,\mathcal{F}_M)$ is the quotient of a foliated manifold $(P,\mathcal{F}_P)$
along a surjective submersion with connected fibers, then the same is true for
the corresponding holonomy groupoids. For quotients by a Lie group action, an
analog statement holds under suitable assumptions, yielding a Lie 2-group
action on the holonomy groupoid.
