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   

-  $\forall K \in [H,G] ,\ \forall g \in G ,\ HgK = KgH$  
- $\sum_i \frac{1}{\vert K_i : H  \vert} \le 2$  

then $\exists V$ irreducible complex representation of $G$ such that $G_{(V^H)} = H$. 


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