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, $G\subseteq Aut(K)$. It is well known that $H^1(G,K^*=1)$ [Hilbert 90]. But does the following hold: $H^1(Aut(K), K^*)=1$ ?

Thanks in advance.