  The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of
the Schur multiplier $H^2(G,\mathbb Q/\mathbb Z)$ consisting of the cohomology
classes which vanish after restricting to every abelian subgroup of $G$. We
give a new proof of a Hopf-type formula for $B_0(G)$ and derive an exact
sequence for the cohomological version of the Bogomolov multiplier. Using this
exact sequence we provide necessary and sufficient conditions for the
corresponding inflation homomorphism to be an epimorphism and to be the zero
map. Finally, we give a complete list of groups of order $p^6$, for odd prime
$p$, having trivial Bogomolov multiplier, so completing the 2020 investigation
of Chen and Ma.
