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

Let $G$ be a finite nilpotent group and $n\in \{3,4,5\}$. Consider $S_n\times G$ as a subgroup of $S_n\times S_{\mathrm{\mid }G\mathrm{\mid }}\subset S_{n\mathrm{\mid }G\mathrm{\mid }}$, where $G$ embeds into the second factor of $S_n\times S_{\mathrm{\mid }G\mathrm{\mid }}$ 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 $S_{n\mathrm{\mid }G\mathrm{\mid }}$. Our result requires that $G$ satisfies some mild conditions.