Let $\GF$ be a Galois field.
Then $\GF$ is perfect.
By Characteristic of Galois Field is Prime, $\Char \GF$ is a prime number, say $p$.
By Frobenius Endomorphism on Field is Injective, $\Frob$ is injective.
By Injection from Finite Set to Itself is Surjection, $\Frob$ is bijective.
By Bijective Ring Homomorphism is Isomorphism, $\Frob$ is an automorphism.