  In this paper we study equivariant algebraic $K$-theory for an action of a
finite constant group scheme in the equivariant motivic homotopy theory. We
prove that the equivariant algebraic $K$-theory is represented by an
equivariant ind-scheme defined by Grassmannians. Using this result we show that
in the category of equivariant motivic spaces the set of endomorphisms of the
motivic space defined by $K_0(G,-)$ coincides with the set of endomorphisms of
infinite Grassmannians in the equivariant motivic homotopy category. To this
end we explicitly recall the folklore computation of equivariant $K$-theory of
Grassmannians.
