  Let $G$ be a locally compact group. For every $G$-flow $X$, one can consider
the stabilizer map $x \mapsto G_x$, from $X$ to the space $\mathrm{Sub}(G)$ of
closed subgroups of $G$. This map is not continuous in general. We prove that
if one passes from $X$ to the universal irreducible extension of $X$, the
stabilizer map becomes continuous. This result provides, in particular, a
common generalization of a theorem of Frol\'ik (that the set of fixed points of
a homeomorphism of an extremally disconnected compact space is open) and a
theorem of Veech (that the action of a locally compact group on its greatest
ambit is free). It also allows to naturally associate to every $G$-flow $X$ a
stabilizer $G$-flow $\mathrm{S}_G(X)$ in the space $\mathrm{Sub}(G)$, which
generalizes the notion of stabilizer uniformly recurrent subgroup associated to
a minimal $G$-flow introduced by Glasner and Weiss.
