On the metacyclic epimorphic images of finite p-groups

We prove that if G is a p-group of order pm > pn, where n > 3 for p = 2 and n > 2 for p > 2, then the number of normal subgroups D of G such that G/D is metacyclic of order pn is a multiple of p, unless G is metacyclic. We also give a very short and elementary proof of the following result: representation groups of nonabelian metacyclic p-groups are metacyclic