  Let $G$ be a group and write $\mbox{Perm}(G)$ for its symmetric group. Define
$\mbox{Hol}(G)$ to be the holomorph of $G$, regarded as a subgroup of
$\mbox{Perm}(G)$, and let $\mbox{NHol}(G)$ denote its normalizer. The quotient
$T(G) = \mbox{NHol}(G)/\mbox{Hol}(G)$ has been computed for various families of
groups $G$, and in most of the known cases, it turns out to be elementary
$2$-abelian, except for two groups of order $16$ and some groups of odd prime
power order and nilpotency class two. In this paper, we shall show that $T(G)$
is elementary $2$-abelian for all finite groups $G$ of squarefree order, and
that $T(G)$ is not a $2$-group for certain finite $p$-groups $G$ of nilpotency
class at most $p-1$.
