I show that when there is a symmetry in a process defined on the nodes of a network, this can be captured by a new structure, the ``group graph'', in which group elements label the links of a network. I show that group graphs are a generalisation of signed networks which are an example of a $Z_2$ group graph. I also show that the concept of balance in signed networks can be generalised to group graphs. Finally, I show how the dynamics of processes on a consistent group graph are completely controlled by the topology of the underlying network, not by the symmetry group. This generalises recent results on signed networks (Tian and Lambiotte, 2024a) and complex networks (Tian and Lambiotte, 2024b).