**Yes.** 
  
This was proved in the [planar algebra][1] framework, see [arXiv:1704.00745][2], Corollary 6.10.    

For a self-contained group-theoretic proof, see [arXiv:1708.02565][3].


  [1]: https://en.wikipedia.org/wiki/Planar_algebra
  [2]: https://arxiv.org/pdf/1704.00745v3.pdf
  [3]: https://arxiv.org/abs/1708.02565