  We prove that the tautological ring of $M_{2,n}^{ct}$, the moduli space of
n-pointed genus two curves of compact type, does not have Poincar\'e duality
for any $n \geq 8$. This result is obtained via a more general study of the
cohomology groups of $M_{2,n}^{ct}$. 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 $H^k(M_{2,n}^{ct})$ for any $k$ and $n$
considered both as $S_n$-representation and as mixed Hodge
structure/$\ell$-adic Galois representation considered up to
semi-simplification. A consequence of our results is also that all even
cohomology of $\overline M_{2,n}$ is tautological for $n < 20$, and that the
tautological ring of $\overline M_{2,n}$ fails to have Poincar\'e duality for
all $n \geq 20$. This improves and simplifies results of the author and Orsola
Tommasi.
