A generalization of convergence actions

Let a group $G$ act properly discontinuously and cocompactly on a locally compact space $X$. A Hausdorff compact space $Z$ that contains $X$ as an open subspace has the perspectivity property if the action $G\curvearrowright X$ extends to an action $G\curvearrowright Z$, by homeomorphisms, such that for every compact $K\subseteq X$ and every element $u$ of the unique uniform structure compatible with the topology of $Z$, the set $\{gK:g\in G\}$ has finitely many non $u$-small sets. We describe a correspondence between the compact spaces with the perspectivity property with respect to $X$ (and the fixed action of $G$ on it) and the compact spaces with the perspectivity property with respect to $G$ (and the left multiplication on itself). This generalizes a similar result for convergence group actions.