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$G/N_{2} \cong N_{1}$impliestheother\mathrm{.}Finite,homocyclic$p$-groupsaremorphic,andsoisthenonabeliangroupoforder$p^{3}$ 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.