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

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 pr 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.