$G \neq gal(E:E_G)$?

Question

Let $E$ be a field and $G$ be a set of automorphism of this field and $E_G$ be the field fixed by these automorphism. I can see that $G \subset gal(E:E_G)$. I was looking for an example to see when the reverse inclusion is not true. Unfortunately, I don't know many simple galois group exept $gal(\mathbb{F}_{p^n}:\mathbb{F}_{p}) \cong C_n$ (cyclic group of order $n$). So I took for instance $gal(\mathbb{F}_{16}:\mathbb{F}_{2}) \cong C_4$. Then a subgroup would be $C_2$. But then is $gal(\mathbb{F}_{16}:(\mathbb{F}_{16})_{C_2}) \neq gal(\mathbb{F}_{16}:\mathbb{F}_{2})?$ Easier examples are welcomed!