In this article the author develops the theory of a Schur multiplier for pairs of groups and he shows that it leads to a more systematic treatment of a number of results on the usual multiplier. In several instances this treatment yields sharper results.
The Schur multiplier of the pair $(G,N)$, where $N$ is a normal subgroup of $G$, is defined as a functorial Abelian group ${\cal M}(G,N)$ whose principal feature is a natural exact sequence $$\multline H_3(G,\bbfZ)\to H_3(G/N,\bbfZ)\to{\cal M}(G,N)\to{\cal M}(G)\to\\ \to{\cal M}(G/N)\to N/[N,G]\to(G)^{ab}\to(G/N)^{ab}\to 0\endmultline$$ (${\cal M}(K)={\cal M}(K,K)$). The definition is given via classifying spaces of groups but a purely homological algebraic definition and proof of the exactness of the above sequence is also given by the author [in Glasg. Math. J. 29, 13-19 (1987;

Zbl 0631.20040)].