**Yes** if the lattice  $[H,G]$ is equal to the boolean lattice $B_n$ with $n < 5$, more precisely:

In [this paper][1] the following theorem is proved: 

> *Theorem:* let $(H \subseteq G)$ be an inclusion of finite groups and $K_1, \dots , K_n$ the minimal overgroups of $H$. If the lattice
> $[H,G]$ is distributive and if one of the following (non-equivalent)
> statements occur   
> 
> -  $\forall K \in [H,G] ,\ \forall g \in G ,\ HgK = KgH$  
> - $\sum_i \vert K_i : H \vert^{-1} \le 2$  
> 
> then $G$ is $H$-linearly primitive.

  

But, $\sum_i \vert K_i : H \vert^{-1} \le n/2$, so by the second point we get a positive answer for $B_n$ with $n < 5$, and then for  $\vert G : H \vert < 32$. By GAP it is checked true for $G$ perfect with $\vert G \vert < 10080$, except $7680$.  


  [1]: http://arxiv.org/abs/1505.06649