  We show that the parametrised topological complexity of Cohen, Farber and
Weinberger gives an invariant of group epimorphisms. We extend various bounds
for the topological complexity of groups to obtain bounds for the parametrised
topological complexity of epimorphisms. Several applications are given,
including an alternative computation of the parametrised topological complexity
of the planar Fadell--Neuwirth fibrations which avoids calculations involving
cup products. We also prove a homotopy invariance result for parametrised
topological complexity of fibrations over different bases.
