We prove that the tautological ring of , the moduli space of -pointed genus two curves of compact type, does not have Poincaré duality for any. This result is obtained via a more general study of the cohomology groups of. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of for any and considered both as -representation and as mixed Hodge structure/ -adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of is tautological for