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.

Let $K/\mathbb F_q(x)$ be a finite Galois extension with Galois group $G$. 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.

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.

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.