We introduce a generalized notion of combinatorial species called A-species, where A is a Hopf algebra. The role played by the symmetric group, Sn, in the classical theory of species is now replaced with the wreath product A≀Sn. We show that category of A-species admits a monoidal structure under the Cauchy and Hadamard product. Under certain choices of A, we recover the notion of Joyal's species, H-species defined by Choquette and Bergeron, and Br-species defined by Henderson. We define many bilax monoidal functors involving the category of A-species. The first functor we construct, SA, goes from the category of species to A-species. This functor sends the regular representation of the symmetric group Sn to the regular representation of A≀Sn, and we use this functor to construct the appropriate definitions of A-Hopf monoids built from common Hopf monoids. When A=KCr, we recover the functor defined by Choquette and Bergeron. We then define A-Fock functors, which are bilax functors between the category of A-species and the category of vector spaces. We analyze the images of certain A-Hopf monoids under them and show that they are all isomorphic as Hopf algebras to the Hopf algebra of polynomials invariant under the hyperoctrahedral group, C⟨⟨x⟩⟩Br. We end by showing a sub Hopf algebra of a quotient of the ring of Cr-colored set partitions functions surjects onto C⟨⟨x⟩⟩Br and has a one-sided inverse.