Extension functions for rank 2, torsion free abelian groups

The set of isomorphism classes of rank 2, torsion free abelian groups with a pure subgroup isomorphic to a given rank 1 group is shown to be in natural 1-1 correspondence with the set of pairs consisting of a quotient type and a type of an extension function. In terms of these invariants, necessary and sufficient conditions are determined for such a group to be homogeneous or to admit a pure cyclic subgroup. Moreover, this 1-1 correspondence has an explicit inverse, so that examples are readily obtained.