In the theory of linear representations of finite groups and their characters, methods for relating characters of a group with characters of one of its subgroups are desirable. In this thesis, we consider a character bijection between characters of a group lying over a fixed character of a normal subgroup, and similar defined characters of a subgroup. This situation occurs in many applications, for example in the proof of character correspondences found by Dade and Isaacs. With the techniques developed here, we can give significantly more transparent proofs of these results. We also get generalizations concerning modular representation theory and derive additional properties of the Isaacs correspondence, in particular that it preserves Schur indices.