On groups well represented as automorphism groups of groups

Assuming Gödel’s axiom of constructibility $\mathbf{V}=\mathbf{L},$ we present a characterization of those groups $L$ for which there exist arbitrarily large groups $H$ such that $aut(H)\cong L$. In particular, we show that it suffices to have one such group $H$ such that the size of its center is bigger than $2^{\mathrm{\mid }L\mathrm{\mid }+{\mathrm{\aleph }}_0}$.