This paper delves into the link between symmetry and controllability of networked systems. We examine symmetry results pertaining to the determining sets of a graph and eigenvalue multiplicity. These provide methods to reason how symmetry structure of the network informs the control input design. We generalize graph automorphisms, which represent graph symmetries, to signed fractional graph automorphisms via a semi-definite programming relaxation. Consequently, we extend existing results on the relationship between graph automorphisms and uncontrollability to signed fractional graph automorphisms, showing necessary and sufficient conditions for controllability.