Finite p-groups of class two with a large multiple holomorph

Let G be any group. The quotient group T(G) of the multiple holomorph by the holomorph of G has been investigated for various families of groups G. In this paper, we shall take G to be a finite p-group of class two for any odd prime p, in which case T(G) may be studied using certain bilinear forms. For any n≥4, we exhibit examples of G of order pn+(n2) such that T(G) contains a subgroup isomorphic to GLn(Fp)×GL(n2)−n(Fp). For finite p-groups G, the prime factors of the order of T(G) which were known so far all came from p(p−1). Our examples show that the order of T(G) can have other prime factors as well. In fact, we can embed any finite group into T(G) for a suitable choice of G.