Let $X$ be a manifold equipped with a proper $G$-action. Suppose we have a central extension $$0\to H\hookrightarrow G'\to G\to 0,$$ where $G\cong G'/H$.
Question 1: Can one construct a manifold $X'$ on which $G'$ acts properly, with the property that $X'/G'$ is homeomorphic to $X/G$?
Question 2: If not, is there a topological obstruction to the existence of $X'$?
In other words, I would like to know: given a class in $\alpha\in H^2(G,H)$ and a proper $G$-manifold $X$, is there a manifold $X'$ on which the extension defined by $\alpha$ acts?
Remark: Since $G'$ is a fiber bundle over $G$ with fiber $H$, my guess is that $X'$ should be a $H$-bundle over $X$, but I'm not sure how to write down $X'$ explicitly.