A Maschke type theorem for relative Hom-Hopf modules

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Homcomodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor F from the category of relative Hom-Hopf modules to the category of right (A, β)-Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the (H, α)-coaction to be separable. This leads to a generalized notion of integrals.