The relative Heller operator and relative cohomology for the Klein 4-group

Let $G$ be the Klein Four-group and let $k$ be an arbitrary field of characteristic 2. A classification of indecomposable $kG$-modules is known. We calculate the relative cohomology groups H_\{chi}^i(G,N) for every indecomposable $kG$-module $N$, where \{chi} is the set of proper subgroups in $G$. This extends work of Pamuk and Yalcin to cohomology with non-trivial coefficients. We also show that all cup products in strictly positive degree in H_\{chi}^*(G,k) are trivial.