Extend an action of indeterminates to an action of field of rational functions

Question

In this text in the theorem 1.15 the author constructs a Galois extension using group action. In his construction, I can see why $G$ can be seen as an automorphism group of $K$, but I cannot see why the elements are $F$-automorphisms. How can I see that the induced action keep fixed the elements of $F$? Because the structure of $F$ is different from $G$, I'm confused about the extended action of $F(T)$.