On Commuting Automorphisms of Some Finite $p$-Groups

Let $G$ be a group and $\operatorname{Aut}(G)$ be its automorphism 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 $A(G)$. We find some cases when $A(G)$ is a subgroup of $\operatorname{Aut}(G)$.