Assumptions: Assume that $G$ is a topological group and $Z_1,Z_2$ are discrete, normal subgroups of $G$ (hence central) and $G / Z_1$ and $G / Z_2$ denote the quotient groups. Assume moreover that there exists a group homomorphism $\varphi \colon G / Z_1 \to G / Z_2$, which is also a topological covering map.
Claim: Then $Z_1 \subseteq Z_2$.
Idea: Of course we know that $\varphi(e \cdot Z_1) = e \cdot Z_2$. But since we have not made any assumptions about $\varphi$ respecting the natural action of $G$ on both spaces, I don't know how to proceed further. Maybe I don't even need that $\varphi$ is a covering map, just the fact that $\varphi$ is surjective.