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]

[![Satz von Mani][2]][2]

My translation:

**Theorem.** *For every graph $\mathfrak{Q}$* there is a three dimensional convex polyhedron $P$ with the properties:

 (a) $\mathfrak{Q}$ is isomorphic to vertex-edge skeleton $\mathfrak{P}^1$ of $P$

 (b) Every automorphism of $\mathfrak{P}^1$ is induced by a symmetry of the polyhedron $P$.

See this [mathscinet link][3] for an english summary of the result. 


  [1]: http://link.springer.com/article/10.1007/BF02075357
  [2]: https://i.sstatic.net/0F2cP.png
  [3]: http://www.ams.org/mathscinet-getitem?mr=296808