Let k be a perfect field of characteristic p>0. We obtain a complete classification of finite abelian local k-Hopf algebras with local dual such that the augmentation ideal is annihilated by the Frobenius map. We then use the theory of finite Honda systems to show that these Hopf algebras lift to extensions R of W(k) with 2≦ e(R/W(k))≦ p-1, and construct all such lifts.