# Galois correspondence of covering spaces of spaces not necessarily semilocally simply-connected

I've been trying to solve the following exercise (1.3.24) from Hatcher's Algebraic Topology:

Given a covering space action of a group $$G$$ on a path-connected, locally path-connected space $$X$$, then each subgroup $$H$$ in $$G$$ determines a composition of covering spaces $$X \to X/H \to X/G$$. Show:

1. Every path-connected covering space between $$X$$ and $$X/G$$ is isomorphic to $$X/H$$ for some subgroup $$H$$ in $$G$$.
2. Two such covering spaces $$X/H_1$$ and $$X/H_2$$ of $$X/G$$ are isomorphic iff $$H_1$$ and $$H_2$$ are conjugate subgroups of $$G$$.
3. The covering space $$X/H \to X/G$$ is normal iff $$H$$ is a normal subgroup of $$G$$, in which case the group of deck transformations of this cover is $$G/H$$.

This is self-study. Not homework.

My thoughts:

I cannot use the main classification theorem since it's only applicable to semilocally simply-connected spaces.

Assume $$X \to Y \to X/G$$. I'm able to show that the group of deck transformations of $$X \to Y$$ (name it $$H$$) is a subgroup of $$G$$. This is because $$G$$ is the group of deck transformations of $$X \to X/G$$ and every isomorphism of $$X \to Y$$ is automatically an isomorphism of $$X \to X/G$$ via covering map composition.

Now I'm stuck. How to show $$Y$$ is isomorphic to $$X/H$$?

I hope the solution to (1) would give me insight to (2) and (3).

I feel that this exercise is analogous to the main classification theorem. This is why I feel I need to solve it, but I am unable to.

Solutions and/or references that cover this question would be welcome.