The Multiple Holomorphs of Finite $p$-Groups of Class Two

$\DeclareMathOperator{\Hol}{Hol}$$\DeclareMathOperator{\Aut}{Aut}$$\newcommand{\Gp}[0]{\mathcal{G}(p)}$$\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$Let $G$ be a group, and $S(G)$ be the group of permutations on the set $G$. The (abstract) holomorph of $G$ is the natural semidirect product $\Aut(G) G$. We will write $\Hol(G)$ for the normalizer of the image in $S(G)$ of the right regular representation of $G$, \begin{equation*} \Hol(G) = N_{S (G)}(\rho(G)) = \Aut(G) \rho(G) \cong \Aut(G) G, \end{equation*} and also refer to it as the holomorph of $G$. More generally, if $N$ is any regular subgroup of $S(G)$, then $N_{S(G)}(N)$ is isomorphic to the holomorph of $N$. G.A.~Miller has shown that the group \begin{equation*} T(G) = N_{S(G)}(\Hol(G))/\Hol(G) \end{equation*} acts regularly on the set of the regular subgroups $N$ of $S(G)$ which are isomorphic to $G$, and have the same holomorph as $G$, in the sense that $N_{S(G)}(N) = \Hol(G)$. If $G$ is non-abelian, inversion on $G$ yields an involution in $T(G)$. Other non-abelian regular subgroups $N$ of $S(G)$ having the same holomorph as $G$ yield (other) involutions in $T(G)$. In the cases studied in the literature, $T(G)$ turns out to be a finite $2$-group, which is often elementary abelian. In this paper we exhibit an example of a finite $p$-group $\Gp$ of class $2$, for $p > 2$ a prime, which is the smallest $p$-group such that $T(\Gp)$ is non-abelian, and not a $2$-group. Moreover, $T(\Gp)$ is not generated by involutions when $p > 3$. More generally, we develop some aspects of a theory of $T(G)$ for $G$ a finite $p$-group of class $2$, for $p > 2$. In particular, we show that for such a group $G$ there is an element of order $p-1$ in $T(G)$, and exhibit examples where $\Size{T(G)} = p - 1$, and others where $T(G)$ contains a large elementary abelian $p$-subgroup.