There has recently been significant interest in studying the impact of symmetry on the infinitesimal rigidity of (bar-joint) frameworks and related geometric constraint systems. A key result is that the rigidity matrix of a symmetric framework can be transformed into a block-decomposed form. For symmetric plane frameworks where the group acts freely on the vertex set, combinatorial characterisations for symmetry-generic infinitesimal rigidity have been established for the reflection group and a class of rotational groups. In this talk, I will show a generalisation of these results to the case where the framework is allowed to have fixed joints