On Commuting Automorphisms of Finite $p$-Groups with a Metacyclic Quotient

Let $G$ be a finite non-Abelian $p$-group, where $p$ is an odd prime, such that $G/Z(G)$ is metacyclic. We prove that all commuting automorphisms of $G$ form a subgroup of $\text{Aut}(G)$ if and only if $G$ is of nilpotence class 2.