In this paper, we define and study an equivariant analogue of Cohen, Farber and Weinberger’s parametrized topological complexity. We show that several results in the non-equivariant case can be extended to the equivariant case. For example, we establish the fibrewise equivariant homotopy invariance of the sequential equivariant parametrized topological complexity. We obtain several bounds on sequential equivariant topological complexity involving the equivariant category. We also obtain the cohomological lower bound and the dimension-connectivity upper bound on the sequential equivariant parametrized topological complexity. In the end, we use these results to compute the sequential equivariant parametrized topological complexity of equivariant Fadell–Neuwirth fibrations and some equivariant fibrations involving generalized projective product spaces.
