  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 $n\geq 4$, we exhibit examples of $G$ of order $p^{n+{n\choose
2}}$ such that $T(G)$ contains a subgroup isomorphic to \begin{equation*}
\operatorname{GL}_n(\mathbb{F}_p) \times
\operatorname{GL}_{\binom{n}{2}-n}(\mathbb{F}_p).\end{equation*} 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 $G$.
