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 . The group structure of N is compatible with the geometric structure of this 2-design, and we call 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 is also a design group. Suppose further that for k=|Φ| we have |N/[N,N]|>2k2−6k+1. We show that and are isomorphic if and only if Φ and Ψ are conjugate in the automorphism group of N.