Topological symmetry groups of graphs embedded in the 3-sphere

The topological symmetry group of a graph embedded in the $3$-sphere is the group consisting of those automorphisms of the graph which are induced by some homeomorphism of the ambient space. We prove strong restrictions on the groups that can occur as the topological symmetry group of some embedded graph. In addition, we characterize the orientation preserving topological symmetry groups of embedded $3$-connected graphs in the $3$-sphere.