Formal groups and lifts of the field of norms

Let $K$ be a finite extension of ${\mathbf{Q}}_p$. The field of norms of a strictly APF extension $K_{\mathrm{\infty }}\mathrm{/}K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{G}\mathrm{a}\mathrm{l}(K_{\mathrm{\infty }}\mathrm{/}K)$. 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 $(\phi ,\mathrm{\Gamma })$-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 $K_{\mathrm{\infty }}\mathrm{/}K$ are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.