  We study a double solid X branched along a nodal sextic surface in a
projective space and the 2-torsion subgroup in the third integer cohomology
group of a resolution of singularities of X. This group can be considered as an
obstruction to rationality of X. Studying this group we conclude that all
sextic double solids admitting non-trivial obstructions to rationality are
branched along determinantal surfaces of very specific type and we provide an
explicit list of them.
