Unique intermediate subgroup and double coset relation I

Question

Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$).

Question: Is it true that $HgK=KgH$, $\forall g \in G$ ?

Experiment (GAP) : It's true if $[G:H] \le 30$ and $\vert G/H_G \vert \le 10000$.
($H_G$ is the normal core of $H$ in $G$)

Answer 1

There are counterexamples with $G = {\rm PSL}(2,11)$, $H$ cyclic of order $6$ and $K$ dihedral of order $12$.

I find it difficult to understand why you would expect such a property to be true. The reason why it holds in many small examples is that $HgK=KgK=KgH$.