In this article we develop the theory of a Schur multiplier for "pairs of groups". The idea of such a multiplier is implicit in the work of J.-L. Loday (1978) and others on algebraic K-theory, and in the work of Eckmann et al. (1972) and others on group homology. In contrast to their work, we focus on the general group-theoretic properties of the multiplier. These properties are systematically derived from: 1) the functoriality of the multiplier; 2) an exact homology sequence; 3) and a transfer homomorphism.