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To smooth schemes equipped with a smooth affine group scheme action, weassociate an equivariant motivic homotopy category. Underlying our constructionis the choice of an `equivariant Nisnevich topology' induced by a complete,regular, and bounded $cd$-structure. We show equivariant $K$-theory of smoothschemes is represented in the equivariant motivic homotopy category. This isused to characterize equivariantly contractible smooth affine curves andequivariant vector bundles on such curves. Generalizations of the purity andblow-up theorems in motivic homotopy theory are shown for actions of finitecyclic groups.

Motivic homotopy of group scheme actions