Continuity of the stabilizer map and irreducible extensions

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.