Let $H$ be a group and $G$ a connected Lie group. Suppose we have a surjective group $f:H \to G$, whose kernel is finite lying in $H$'s centre $Z(H)$. The question is, how to show that $H$ has a unique Lie group structure such that $f$ is also a Lie group homomorphism, as well as a covering map.
My attempt was identifying $V_i = f^{-1}(U_i)$ and $\psi_i = \varphi_i \circ f$ for the atlas of $H$ while $(U_i,\psi_i)_i$ is an atlas for $G$. But I think my attempt solved nothing at all. Perhaps I had only used surjectivity of $f$, meanwhile I guess the finiteness of its kernel, and the connectedness of $G$, should have been used already.
Would you like to give any hint or a proof for this question? I found no document teaching this on the internet :(