Let $k$ be an algebraically closed field of characteristic $p>0$. Let $R$ be a finitely generated (possibly non-reduced) $k$-algebra. Let $\mathbf{S} = \mathrm{Spec} (R)$ be the affine scheme of finite type over $k$ and let $\mathbf{S}(k)$ denote the set of all $k$-valued points of $\mathbf{S}$. If $\mathbf{S}(k)$ has an abstract group structure, namely, a morphism of schemes $\mathbf{S}(k) \times \mathbf{S}(k) \to \mathbf{S}(k)$ satisfying the usual group axioms, my question is that
Is there a natural way to give $\mathbf{S}$ a group scheme structure (namely, give $R$ a Hopf algebra structure) $\mathbf{S} \times \mathbf{S} \to \mathbf{S}$ such that it induces the same multiplication map at the $k$-valued points?