This paper explores the role of symmetry when establishing output controllability and stabilizability of LTI networked systems. Previous work, which examined graph symmetry through its signed fractional graph automorphisms, has been extended to output controllability and stabilizability. Conditions on input symmetry, output symmetry, and non-output symmetry over the network are proved to provide necessary and sufficient conditions for output controllability. Results pertaining to output stabilizability provide insight into stabilization of negatively weighted consensus dynamics. The approach also provides a semi-definite programming formulation of output controllability and stabilizability.