Formal groups and lifts of the field of norms

Let K be a finite extension of Qp. The field of norms of a strictly APF extension K∞/K is a local field of characteristic p equipped with an action of Gal(K∞/K). When can we lift this action to characteristic zero, along with a compatible Frobenius map? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of (φ, Ɣ)-modules, and show that under a certain assumption on the type of lift, such an extension is generated by the torsion points of a relative Lubin–Tate group and that the power series giving the lift of the action of the Galois group of K∞/K are twists of semiconjugates of endomorphisms of the same relative Lubin–Tate group.