Autocommutator subgroups of finite groups (Q1355621)

Let \(G\) be a group. The absolute centre \(L(G)\) is the set of all elements of \(G\) which are fixed by every automorphism of \(G\), while the autocommutator subgroup \(G^*\) of \(G\) is the subgroup generated by all the elements of the form \(g^{-1}g^\alpha\), where \(g\) is in \(G\) and \(\alpha\) is any automorphism of \(G\). The author shows that if \(K\) is any finite group, then there are only finitely many finite groups \(G\) for which \(G/L(G)\) is isomorphic with \(K\), and finitely many groups \(G\) for which \(G^*\) is isomorphic with \(K\).