Finite morphic p-groups

According to Li, Nicholson and Zan, a group $G$ is said to be morphic if, for every pair $N_1, N_2$ of normal subgroups, each of the conditions $G/N_1 \cong N_2$ and $G/N_2 \cong N_1$ implies the other. Finite, homocyclic $p$-groups are morphic, and so is the nonabelian group of order $p^3$ and exponent $p$, for $p$ an odd prime. It follows from results of An, Ding and Zhan on self dual groups that these are the only examples of finite, morphic $p$-groups. In this paper we obtain the same result under a weaker hypotesis.