If $L/K$ is a field extension, is $\text{Aut}(K)$ a normal subgroup of $\text{Aut}(L)$ (possibly under some extra conditions on $L$ and $K$) and if so what is the quotient isomorphic to, possibly $\text{Aut}(L/K)$, if this is at all correct?
Note: This is when thinking functions in $\text{Aut}(K)$ map $L\setminus K$ with the identity.