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 $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity)
Recall that $(H \subset K) \sim (K \subset G)$ if $(H/H_K \subset K/H_K) \simeq (K/K_G \subset G/K_G)$, with $H_K$ the normal core (in others words, these inclusions define the same transitive permutation group).
Remark: $G$ can't be a finite simple group. In fact $K/H_K \simeq G/K_G$ so if $G$ is a finite simple group, then $K_G=\{ e \}$, and so $\vert G \vert \le \vert K \vert$, contradiction.
Examples: Up to $\sim$, there are exactly $123$ such inclusions of groups of index $\le 30$.
The first examples are $(\{e\} \subset \mathbb{Z}_{p^2})$
and $(\mathbb{Z}_2 \subset D_{p^2} )$.
Question: Is it true that $HgK=KgH$, $\forall g \in G$ ?
Experiment (GAP): It's checked for $\vert G \vert \le 2000$, except $1024$ and $1536$.
It's also true if $[G:H] \le 30$ (see this comment of Derek Holt).
