Bounds for the index of the centre in capable groups

A group $H$ is called capable if it is isomorphic to $G/\mathbf{Z}(G)$for some group $G$. Let $H$ be a capable group. I. M. Isaacs (2001) showed that if $H$ is finite, then the index of the centre is bounded above by some function of $\vert H'\vert$. We show that if $\vert H'\vert<\infty$, then $\vert H:Z(H)\vert\leq \vert H'\vert^{c\log_2\vert H'\vert}$ with some constant $c$ and this bound is essentially best possible. We complete a result of Isaacs, showing that if $H'$ is a cyclic group, then $\vert H:\mathbf{Z}(H)\vert\leq \vert H'\vert^2$.