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?