On commuting automorphisms of finite groups (Q2281555)

For a finite group \(G\), let \(A(G)\) be the set of those automorphisms \(\alpha\) of \(G\) having the property that \([x, \alpha (x)]=1\) for every \(x\in G\). The set \(A(G)\) is not always a subgroup of \(\Aut(G)\) and the author shows in this paper that \(A(G)\) is a subgroup of \(\Aut(G)\) provided that \(G\) is either: 1) A finite group with cyclic Sylow \(p\)-subgroups for all primes \(p\) dividing the order of \(G\). 2) G is isomorphic to \(\mathrm{GL}(n, q)\) for \(n=3\) or \(q>n\), or \(G\) is isomorphic to \(\mathrm{PSL}(2, q)\). 3) \(G=H\times K\), where \(H\) and \(K\) have no common direct factor and moreover \(A(H)\leqslant\Aut(H)\) and \(A(K)\leqslant \Aut(K)\).