Let $L_2/L_1/K$ be a tower of field extensions where $L_2/K$ is finite algebraic [so $L_1/K$ is also finite]. Prove or disprove: For every $\sigma_1\in Aut_K(L_1)$ there is a $\sigma_2\in Aut_K(L_2)$ such that $\sigma_2|L_1=\sigma_1$.
The claim is clearly true if $L_2/K$ is normal, since we can extend $\sigma_1$ to a homomorphism into an algebraic closure and then use normality. Other than that, I'm completely lost. Do I need to induct on the degree of the extension? (Hints are appreciated.)