Finite morphic p-groups

According to Li, Nicholson and Zan (2010), a group G is said to be morphic if, for every pair N1,N2N1,N2 of normal subgroups, each of the conditions G/N1≅N2G/N1≅N2 and G/N2≅N1G/N2≅N1 implies the other. Finite, homocyclic p -groups are morphic, and so is the nonabelian group of order p3p3 and exponent p, for p an odd prime. It follows from results of An, Ding and Zhan (2011) 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 hypothesis. Both authors are members of GNSAGA - INdAM.