An exact sequence and triviality of Bogomolov multiplier of groups

The Bogomolov multiplier $B_0(G)$ of a finite group $G$ is the subgroup of the Schur multiplier $H^2(G,\mathbb{Q}\mathrm{/}\mathbb{Z})$ 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.