Non-rational sextic double solids

In 1971 Artin and Mumford has constructed an example of non-rational unirational variety. Namely they showed that the torsion subgroup in $H^3(X,Z)$ is a birational invariant for a smooth variety $X$ and described a nodal quartic double solid such that its resolution of singularities admits a non-trivial 2-torsion subgroup in third cohomologies. I will tell about nodal sextic double solids, describe four families of such varieties with non-trivial obstruction of Artin-Mumford type for a general element in each family and show that there is no sextic double solid with non-trivial obstruction outside these families.