Given a group $G$, its centre $Z(G)$ is a normal subgroup of $G$ and one can consider the quotient $G/Z(G)$.

Which groups $H$ can occur as $G/Z(G)$ for some group $G$?

For example, it is easily proved that $G/Z(G)$ cannot be cyclic unless it is trivial.

A certain amount of general literature survey and google search did not give me any answer. But since I do not work in algebra, it could be the case that I missed something easy.

It can be shown that $G/Z(G)$ is isomorphic to the group of inner automorphisms $\mathrm{Inn}(G)$ of $G$.