A group G is called capable if it is the group of inner automorphisms of some group E. Capable pairs are defined in terms of a relative central extension. In this paper we introduce the precise center for a pair of groups and prove that this subgroup makes a criterion for characterizing the capability of the pair. We also show that our result sharpens the obtained result in this area. A complete classification of finitely generated abelian capable pairs will also be given.

Keywords