generalized baer-invariant of a pair of groups and the direct limit

in this paper we introduce the concept of generalized baer-invariant of a pair of groups with respect to two varieties ? and ? of groups. we give some inequalities for the generalized baer-invariant of a pair of finite groups, when ? is considered to be the schur-baer variety. further, we present a sufficient condition under which the order of the generalized baer-invariant of a pair of finite groups divides the order of the generalized baer-invariant of a pair of their factor groups. moreover, we prove that the generalized baer-invariant of a pair of groups commutes with direct limit.