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