For every finite group G, we introduce the moduli space of pointed, admissible G-covers, a space which reduces to the moduli space of stable curves when G is the trivial group. We show that its associated equivariant cohomological field theories possess a kind of equivariant Frobenius algebra structure studied by Turaev. We introduce an appropriate "quotient" for such theories and show that this procedure yields a cohomological field theory in the sense of Kontsevich-Manin. This process can be regarded as a kind of orbifolding procedure for such theories. Those theories associated to orbifolds which are global quotients by G give rise to the Fantechi-Gottsche ring and its quotient, the Chen-Ruan orbifold cohomology.