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.