Prove that $\operatorname{Gal}(K_1 / F_1)$ and $\operatorname{Gal}(K_2 / F_2) $ are isomorphic

Question

Let $K_1/F_1$ a field extension, and $\sigma : K_1 \rightarrow K_2$ a field isomorphism.

If $F_2=\sigma(F_1)$, prove that $\operatorname{Gal}(K_1 / F_1)$ and $\operatorname{Gal}(K_2 / F_2)$ are isomorphic.

Intuitively, I see that for any $\phi \in\operatorname{Gal}(K_1/F_1)$, $\sigma \circ \phi \circ \sigma^{-1}$ is an automorphism that fix any $a\in F_2$, hence $\sigma \circ \phi \circ \sigma^{-1} \in\operatorname{Gal}(K_2 / F_2)$.

But I don't know if it is enough to prove the statement.