Automorphisms of certain design groups II

Let Φ be a group acting semiregularly as automorphisms on the group (N,+). This gives rise to a certain 2-design (N,BΦ). The group structure of N is compatible with the geometric structure of this 2-design, and we call (N,BΦ,+) a design group. Extending our previous results, we study the question of when such design groups are isomorphic. Let Ψ be another group acting on N so that (N,BΨ,+) is also a design group. Suppose further that for k=Φ we have N/[N,N]>2k2−6k+1. We show that (N,BΦ,+) and (N,BΨ,+) are isomorphic if and only if Φ and Ψ are conjugate in the automorphism group of N.