  In this note, we consider the notion of what we call \emph{recognizing
spaces} for stable subgroups of a given group. When a group $G$ is a mapping
class group or right-angled Artin group, it is known that a subgroup is stable
exactly when the largest acylindrical action $G \curvearrowright X$ provides a
quasi-isometric embedding of the subgroup into $X$ via the orbit map. In this
sense the largest acylindrical action for mapping class groups and right-angled
Artin groups provides a recognizing space for all stable subgroups. In
contrast, we construct an acylindrically hyperbolic group (relatively
hyperbolic, in fact) whose largest acylindrical action does not recognize all
stable subgroups.
