Let $\mathbb{F}_p$ be the finite field with $p$ (prime) elements. What is the group of (continuous) field automorphisms $$ \text{Aut}(\mathbb{F}_p((t)))? $$ If I am not mistaken, then $$ \text{Aut}(\mathbb{F}_p[[t]]) $$ should consist of those maps $\varphi$ given by $$ \varphi(t) \in t \cdot \mathbb{F}^{\times}_p + t^2 \cdot \mathbb{F}_p[[t]] $$ or something like that. Question is, how big can the former become?
Comment: It is not really something that I need for my research, it is just a curiosity.