Bogomolov multiplier, double class-preserving automorphisms and modular invariants for orbifolds

We describe the group of braided tensor autoequivalences of the Drinfeld centre of a finite group $G$ isomorphic to the identity functor (just as a functor) as a semi-direct product $Aut^1_{br}(\Z(G))\ \simeq\ Out_{2-cl}(G)\ltimes B(G)\ $ of the group of double class preserving automorphisms and the Bogomolov multiplier of $G$. The Bogomolov multiplier $B(G)$ is the subgroup of its Schur multiplier $H^2(G,k^*)$ of classes vanishing on abelian subgroups of $G$. We show that elements of $Aut^1_{br}(\Z(G))$ give rise to different realisations of the charge conjugation modular invariant for $G$-orbifolds of holomorphic conformal field theories.