  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.
