  We extend the theory of tautological classes on moduli spaces of stable
curves to the more general setting of moduli spaces of admissible Galois covers
of curves, introducing the so-called H-tautological ring. The main new feature
is the existence of restriction-corestriction morphisms remembering
intermediate quotients of Galois covers, which are a rich source of new
classes. In particular, our new framework includes classes of Harris-Mumford
admissible covers on moduli spaces of curves, which are known in some (and
speculatively many more) examples to lie outside the usual tautological ring.
We give additive generators for the H-tautological ring and show that their
intersections may be algorithmically computed, building on work of Schmitt-van
Zelm. As an application, we give a method for computing integrals of
Harris-Mumford loci against tautological classes of complementary dimension,
recovering and giving a mild generalization of a recent quasi-modularity result
of the author for covers of elliptic curves.
