Let \(A(G)=\{\alpha\in\Aut(G)\mid x\alpha(x)=\alpha(x)x\) for all \(x\) in \(G\}\) and \(\text{Cent}(G)=\{\alpha\in\Aut(G)\mid\alpha(C(x))=C(x)\) for all \(x\) in \(G\}\). The authors show that \(A(G)\) is not necessarily a subgroup of \(\Aut(G)\) but \(\alpha^2\in\text{Cent}(G)\) for all \(\alpha\in A(G)\) (Lemma 2.4(i)); if \(G\) is Noetherian, then \([G,A(G)]\leq Z_\infty(G)\) (Theorem 1.1); if \(G\) is finite, there is a nilpotent subgroup \(H\) such that the map restricting \(\alpha\in A(G)\) to \(H\) is injective (Theorem 1.2), and \([G^2,A(G)]\leq Z_2(G)\) (Theorem 1.4).