  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.
