Central quotient of $G=K\rtimes H$ where $H$ is core-free and $Z(G)\leqslant K$

Question

If $G=K\rtimes H$ is a semidiect product of a core-free group $H$ and a normal subgroup $K$ such that $Z(G)\leqslant K$, what can we conclude about the quotient $G/Z(G)$?