On Malle's conjecture for the product of symmetric and nilpotent groups

Abstract: Let $G$ be a finite nilpotent group and $n\in {3,4, 5}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{|G|}\subset S_{n|G|}$, where $G$ embeds into the second factor of $S_n\times S_{|G|}$ via the regular representation. Over any number field $k$, we prove the strong form of Malle's conjecture for $S_n\times G$ viewed as a subgroup of $S_{n|G|}$. Our result requires that $G$ satisfies some mild conditions.