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 p n+( n 2 ) such that T (G) contains a subgroup isomorphic to GL n (F p ) × GL ( n2 )−n (F p ). For finite p-groups G, the prime factors of the order of T (G) which are 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.

We consider the quotient group T (G) of the multiple holomorph by the holomorph of a finite p-group G of class two for an odd prime p. By work of the first-named author, we know that T (G) contains a cyclic subgroup of order p r−1 (p − 1), where p r is the exponent of the quotient of G by its center. In this paper, we shall exhibit examples of G (with r = 1) such that T (G) has order exactly p − 1, which is as small as possible.