Commutators and commutator subgroups of finite $p$-groups

We present a classification of finite $p$-groups $G$, $p \ge 3$, with $\gamma_2(G)$, the commutator subgroup of $G$, of order $p^4$ and exponent $p$ such that each element of $\gamma_2(G)$ is a commutator. As a consequence, it follows that for any $p$-group $G$ of order at least $p^9$, $p \ge 5$, with $|\gamma_2(G)| = p^4$, every element of $\gamma_2(G)$ is a commutator.