  The Bogomolov multiplier of a finite group $G$ is defined as the subgroup of
the Schur multiplier consisting of the cohomology classes vanishing after
restriction to all abelian subgroups of $G$. This invariant of $G$ plays an
important role in birational geometry of quotient spaces $V/G$. We show that in
many cases the vanishing of the Bogomolov multiplier is guaranteed by the
rigidity of $G$ in the sense that it has no outer class-preserving
automorphisms.
