Propagating sharp group homology decompositions

A collection C of subgroups of a finite group G can give rise to three different standard formulas for the cohomology of G in terms of either the subgroups in C or their centralizers or their normalizers. We give a short but systematic study of the relationship among such formulas for nine standard collections C of p-subgroups, obtaining some new formulas in the process. To do this, we exhibit some sufficient conditions on the poset C which imply comparison results.