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 $G$.
Best Answer
In the literature the following terminology is common. If $H$ is a group such that $H \cong G/Z(G)$ for some $G$, we say that $H$ is capable. As you mention, non-trivial cyclic groups are not capable. Another example is the quaternion group of order $8$, which is not capable.
The earliest paper on this seems to be the one by Baer: "Groups with preassigned central and central quotient group. Trans. Amer. Math. Soc. 44 (1938), no. 3, 387–412." He gives a complete classification of finite abelian $H$ that are capable.
In general the description of groups that are isomorphic to some $G/Z(G)$ is not known and seems to be a difficult problem. But you will find many papers on the subject for example by looking up "capable groups" on google, or by looking up on MathSciNet which papers cite the 1938 paper of Baer.