On Insoluble Transitive Subgroups in the Holomorph of a Finite Soluble Group

A question of interest both in Hopf-Galois theory and in the theory of skew braces is whether the holomorph $\mathrm{Hol(N)}$ of a finite soluble group $N$ can contain an insoluble regular subgroup. We investigate the more general problem of finding an insoluble transitive subgroup $G$ in $\mathrm{Hol}(N)$ with soluble point stabilisers. We call such a pair $(G,N)$ irreducible if we cannot pass to proper non-trivial quotients $\overline{G}$, $\overline{N}$ of $G$, $N$ so that $\overline{G}$ becomes a subgroup of $\mathrm{Hol}(\overline{N})$. We classify all irreducible solutions $(G,N)$ of this problem, showing in particular that every non-abelian composition factor of $G$ is isomorphic to the simple group of order $168$. Moreover, every maximal normal subgroup of $N$ has index $2$.