Let $K/\mathbb F_q(x)$ be a finite Galoisian extension with Galois group. Let $Aut(K)$ be the group of $\mathbb F_q$-automorphisms of $K$. Obviously, $G\subseteq Aut(K)$.

Do we have the following: $H^1(Aut(K), K^*)=1$ ?

Thanks in advance.