We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopica
l purity and blow-up theorems for finite abelian groups.
We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of e
quivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
