I will report on joint work with Zhizhang Xie and Guoliang Yu on functoriality properties of the higher index and higher rho invariant for elliptic differential operators on manifolds with symmetry. We show that given a homomorphism between the deck transformation group of a Galois cover and a quotient by a normal subgroup, the map induced on the level of group C*-algebras naturally relates the higher indices and higher rho invariants associated to the two group actions. We work with the maximal version of the group C*-algebra. Our results can be applied to the problem of computing higher invariants on a covering space, for example by relating it to higher invariants on finite-sheeted covers.