Algebraic models for equivariant rational homotopy theory for discrete groups

We provide a framework which generalizes algebraic models of a homotopy theory of spaces to the genuine equivariant case for a discrete group. We explain how this applies to commutative differential graded algebra (cdga) models and complete differential graded Lie algebra models for rational spaces. We compare the cdga model to other model categories in the literature, correcting some mistakes in previous approaches. As an intermediary result, we get that the injective model structure on the category of non-negative cochain complexes with values in a Grothendieck abelian category is cofibrantly generated, and we provide explicit generating (acyclic) cofibrations.