Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph.
A symmetry of polyhedral graph means that a vertex permutation( where the corresponding edges/faces permutations are also induced from vertices) preserving the combinatorial type. We can easily check that this is realized as a planar isotopy and reflection on 2-sphere. I want to say this is combinatorial symmetry and consider the group of combinatorial symmetries for a polyhedral graph.
Now, my question is the following.
Let us consider a polyhedral graph and the combinatorial symmetry group. Is there a Euclidean convex polyhedron of the polyhedral graph whose combinatorial symmetry group can be realized as Euclidean isometries?
For example, $K_4$ has the tetrahedral symmetry group. If one consider regular tetrahedra in Euclidean space, the all tetrahedral symmetries can be realized as Euclidean isometries.
Is this true for all polyhedral graph? .