  In this manuscript, a solution to Problem 18.91(b) in the Kourovka Notebook
is given by proving the following theorem. Let $P$ be a Sylow $p$-subgroup of a
group $G$ with $|P| = p^n$. Suppose that there is an integer $k$ such that $1 <
k < n$ and every subgroup of $P$ of order $p^k$ is $S$-propermutable in $G$,
and also, in the case that $p=2$, $k = 1$ and $P$ is non-abelian, every cyclic
subgroup of $P$ of order $4$ is $S$-propermutable in $G$. Then $G$ is
$p$-nilpotent.
