Torsion of simple abelian group extension

Question

I'm curious if there is a general way to determine the torsion of a group extension. I'm most interested in the simple example where we have a central extension $$ 1 \to \mathbb Z \xrightarrow{f} G \xrightarrow{g} \mathbb Z / n\mathbb Z \to 1,$$ where it is known that the map $f$ acts as multiplication by some integer. What are the possible degrees of torsion elements in $G$? Furthermore, given the map $f$, is the torsion of $G$ determined? For example, if $f$ acts as multiplication by $n$, then $G$ must be $\mathbb Z$. On the other hand, if $f$ acts as multiplication by 1, then we have $G = \mathbb Z \oplus \mathbb Z/ n\mathbb Z$. Are there some simple general statements that can be made here?