  In this note we define a subgroup $H^i_{nr,\pi}$ of unramified cohomology
group $H^i_{nr}$ of a fibration $\pi:X\to S$. This subgroup can be used
efficiently in refined specialization arguments and allows to detect the
failure of stable rationality of a variety specializing to $X$. We compute
$H^2_{nr, \pi}$ systematically for many cubic surface bundles $\pi:X\to S$ over
a smooth projective rational surface over an algebraically closed field: we
give a combinatorial formula in terms of components of the discriminant divisor
of $\pi$.
