Is there a element of absolute Galois group of $\mathbb{Q}$ that is an extension of some 'easy map' like $\sigma : \mathbb{Q}(\sqrt{2}) \rightarrow \mathbb{Q}(\sqrt{2})$, $\sigma(x+y\sqrt2)=x-y\sqrt2$.
What I mean is, for any algebraic extension of $\mathbb{Q}$, say $L$, and any $\mathbb{Q}$-embedding of $L$, say $\sigma$, is there a $\mathbb{Q}$-embedding of $\mathbb{\overline{Q}}$ whose restriction to $L$ is $\sigma$?
(I know the case $L=\mathbb{Q}(i)$ and the conjugation.)