Natural mappings of relative Wall groups

The author studies the natural mappings of the relative Wall groups for the imbedding $\mathbf{Z}\pi\to\widehat{\mathbf{Z}}_2\pi$ that occur in a two-row diagram, in the case of finite abelian 2-groups. It is shown that in this case the diagram splits into a direct sum of a certain number of four different diagrams. The results are applied to the study of mappings of surgery obstruction groups.