Yes, this is true for all polyhedral graphs. See the following reference:

[P. Mani, *Automorphismen von polyedrischen Graphen*, Mathematische Annalen Volume 192, Issue 4, pp 279-303, August 1971][1]

[![enter image description here][2]][2]

  [1]: http://link.springer.com/article/10.1007/BF02075357
  [2]: https://i.sstatic.net/0F2cP.png