Indecomposable symplectic $k(C_2\times C_2)$--modules and their quadratic forms

For the Klein-Four Group $G$ and a perfect field $k$ of characteristic two we determine all indecomposable symplectic $kG$-modules, that is, $kG$-modules with a symplectic, $G$-invariant form which do not decompose into smaller such modules, and classify them up to isometry. Also we determine all quadratic forms that have one of the above symplectic forms as their associated bilinear form and describe their isometry classes.