Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic

We extend the results of G.~Garkusha and I.~Panin on framed motives of algebraic varieties [4] to the case of a finite base field, and extend the computation of the zeroth cohomology group $H^0(\mathbb ZF(\Delta^\bullet_k,\mathbf G^{\wedge n}_m))=K^{MW}_n$ by A.~Neshitov [8] to perfect fields $k$ of positive characteristic different from 2.