Edit: from now on please assume $H=SL(2,\mathbb R), X=PSL(2,\mathbb R)$
I'm dealing with a topological group (assume some "good" group if needed, perhaps a lie group. But do not assume compact group, despite the fact that it should make everything here rather easy) $G$ which is locally isomorphic to some group $H$. Say I can find a covering map $p:H\to X$ for some topological space $X$.Can I derive a covering map $q:G\to X$?
The idea that makes it reasonable for me is that for every $x\in G$ we can take a neighborhood $U_x$ which is isomorphic to $H$ and define $q|_{U_x}=p$. I think that still for every $x\in X$ the pre-image would satisfy the demands of a covering map, but the main problem with my idea is the well-defined and continuity of $q$.
Another idea is to assume local compactness of $G$ (which is fine by me), and maybe (I can't see that clearly) that $G$ is a finite disjoint union of copies of $H$ (which seems quite strong and probably wrong).