Product of subgroups H,K - does HKH=KHK if HKH is a subgroup? Suppose $H,K$ are subgroups of a group $G$ such that $HKH$ is a subgroup of $G$. Is it then true that $KHK=HKH$?

A counterexample is $G=A_4$ with $|H|=3$, $|K|=2$, $HKH=G$, $|KHK|=10$.
More generally, let $G$ be any (finite) $2$-transitive group, $H$ a point stabilizer and $K$ of order $2$ with $K \not\subseteq H$. Then $HKH=G$ but, if the degree is at least $5$, then $|KHK| \le 4|H| < |G|$.
