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