Is this a subgroup of $H$ ?

Question

We have that $F\leq K\leq N$ and

$G=\text{Gal}(N/F)=\{\sigma \in G : \sigma (x)=x, \forall x\in F\} \\ H=\text{Gal}(N/K)=\{\sigma \in G : \sigma (x)=x, \forall x\in K\}$

Does it hold that $$\bigcap_{\sigma \in G}\sigma H\sigma^{-1}\leq H$$ ?

Answer 1

Yes, it is always true, not only in Galois groups. The core of $\;H\;$ , that intersection you wrote, is characterized by being a subgroup of $\;G\;$ which is maximal wrt being normal in $\;G\;$ and contained in $\;H\;$