Suppose $P$ is a finite $p$-group with center $Z(P)$ of order $p$. What kind of groups can appear as the central quotient $P/Z(P)$?
For example, the quotient is in particular a so-called capable group (a group $G$ is called capable if there is a group $H$ such that $H/Z(H)$ is isomorphic to $G$). For example, the quaternion group is not capable and thus cannot appear.
I am looking for some necessary conditions rather than a complete answer (and I doubt that giving a complete answer is possible). What properties does $P/Z(P)$ have? I am mostly interested in the case that $p = 2$.