  We show that the group cohomology of the diffeomorphisms of the disk with $n$
punctures has the cohomology of the braid group of $n$ strands as the summand.
As an application of this method, we also prove that there is no cohomological
obstruction to lifting the "standard" embedding
$\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism
between diffeomorphism groups.
