On reconstructing separable reduced p-groups with a given socle

Let {Mathematical expression} be a separable reduced (abelian)p-group which is torsion complete. We ask whether for {Mathematical expression} there is {Mathematical expression}, H[p]=G[p], H not isomorphic to G. If G is the sum of cyclic groups or is torsion complete, the answer is easily no. For other G, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character of G is equal to Min _n<ω|p ⁿG|, i.e., {Mathematical expression} Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.