  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.
