I define a "group graph" which encodes the symmetry in a dynamical process on a network. Group graphs extend signed networks, where links are labelled with plus or minus one, by allowing link labels from any group and generalising the standard notion of balance. I show that for processes on a balanced group graph the time evolution is completely determined by the network topology, not by the group structure. This unifies and extends recent findings on signed networks (Tian \& Lambiotte, 2024a) and complex networks (Tian \& Lambiotte, 2024b). I will also relate the results discussed here to related work such as the "group graph" of Harary (1982), a "voltage graph" (Gross, 1974) and a "gain graph" (Zaslavsky 1989). Finally, I will review some promising applications for network dynamics and symmetry-driven modelling including status, edges with a zero label, weak balance, unbalanced group graphs and using monoids.