20.108 (2022)

The holomorph $\text{Hol}(G)$ of a group $G$ can be defined as the normalizer of the subgroup of left translations in the group of all permutations of the set $G$. The multiple holomorph $\text{NHol}(G)$ of $G$ is the normalizer of the holomorph. Set $T(G) = \text{NHol}(G)/\text{Hol}(G)$.
$\qquad$ a) Is there a centerless group $G$ for which $T(G)$ is not an elementary abelian 2-group?
$\qquad$ b) Is there a finite centerless group $G$ for which $T(G)$ is not an elementary abelian 2-group?
$\qquad$ c) (A. Caranti). Is there a finite $p$-group $G$ for which the order of $T(G)$ has a prime divisor not dividing $(p-1)p$?