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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 ofgroups $G$, 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 primepower 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, andthat $T(G)$ is not a $2$-group for certain finite $p$-groups $G$ of nilpotencyclass at most $p-1$.

On the multiple holomorph of groups of squarefree or odd prime power order