For (a), Since X is a path-connected, locally path-connected space and the action of G is a covering space action,

is a subgroup of

. By using a

Galois's correspondence, there exists a covering space X/H of X/G such that

is a subgroup of

. In a similar vein, X is the covering space of X/H.

For (b), an isomorphism between covering spaces

and

is a homeomorphism

such that

. This implies that for each

, f preserves the covering space structure, taking

to

.

If H_1 and H_2 are conjugate subgroups of G, then so are

and

in

, where

and

. The remaining step is to show that there exists a bijection between each point

and

for each point

.

For (c), if

is normal, then

is a normal subgroup of

. The group of deck transformation is N(L)/L (Hatcher p71, proposition 1.39b). Since L is a normal subgroup of

, we have

. It remains to show that

. This equation intuitively makes sense, but I can't think of the rigorous proof for now. Can you proceed from here?