Height of Some Automorphisms of Local Fields

In this note, we determine which automorphism subgroups of ${\mathrm{A}\mathrm{u}\mathrm{t}}_{{\mathbb{F}}_q}({\mathbb{F}}_q((x)))$ are corresponding to ${\mathbb{Z}}_p$-extensions or ${\mathbb{Z}}_p\times {\mathbb{Z}}_p$-extensions of characteristic $0$ fields.