On Commuting Automorphisms of a Finite Nonabelian Group with Metacyclic Central Quotient

Let $$G$$ be a group. An automorphism $$\alpha$$ of $$G$$ is called a commuting automorphism if $$\alpha(x)x= x \alpha(x)$$ for all $$x \in G$$ . The set of all commuting automorphisms of $$G$$ is denoted by $$A(G)$$ . The set $$A(G)$$ does not necessarily form a subgroup of the automorphism group of $$G$$ . If $$A(G)$$ is a subgroup of the automorphism group of $$G$$ , then we say that $$G$$ is an $$A$$ -group. Garg [Math. Notes, 106(2), 296–298] stated that if $$G$$ is a finite nonabelian $$p$$ -group (where $$p$$ is an odd prime) such that $$G/Z(G)$$ is a metacyclic group, then $$G$$ is an $$A$$ -group if and only if $$G$$ is of nilpotency class $$2$$ . We identify that this statement is flawed and provide the correct statement. Moreover, we generalize the result by proving that if $$G$$ is a finite nonabelian group such that $$G/Z(G)$$ is a metacyclic group, then $$G$$ is an $$A$$ -group.