Fourth Isomorphism Theorem: Let $L$ be a group and $K$ a normal subgroup of $L$. Then, there is a bijection between the set of groups $H$ with $K\leq H \leq L$ and the subgroups of $L/K$.
Galois Correspondence: Let $L/K$ be a field extension with Galois Group $\text{Aut}(L/K)$. Then, there is a bijection between the set of fields $H$ such that $K\subset H\subset L$ and the subgroups of $\text{Aut}(L/K)$.
I recognise also that the Fourth Isomorphism Theorem can also be stated for rings, and if this is more related, then please use at will. These theorems can be stated very similarly as seen above. Is this the reason for the $L/K$ notation of field extensions (as I have been very confused why the same notation has been chosen)? Does Galois Correspondence imply the Fourth Isomorphism Theorem? If not, what are the links between the two theorems?