The permutation dimension of the Klein 4-group

Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. Balmer and Gallauer's recent result on finite $p$-permutation resolutions of $kG$-modules motivates the study of an intriguing new invariant; the $p$-permutation dimension. Following Walsh's success with cyclic groups of prime order, we compute the (global) $p$-permutation dimension of the Klein 4-group in characteristic $p := 2$, along with the dimensions for each of its indecomposable modules.