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

We describe the group Aut₁ₑ¹ (Z (G) ) Autbr1 (Z (G) ) of braided tensor autoequivalences of the Drinfeld centre of a finite group G isomorphic to the identity functor (just as a functor). We prove that the semi-direct product Out2 − cl (G) ⋉B (G) of the group of double class preserving automorphisms and the Bogomolov multiplier of G is a subgroup of Aut₁ₑ¹ (Z (G) ) Autbr1 (Z (G) ). An automorphism of G is double class preserving if it preserves conjugacy classes of pairs of commuting elements in G. The Bogomolov multiplier B (G) is the subgroup of its Schur multiplier H2 (G, k*) of classes vanishing on abelian subgroups of G. We show that elements of Aut¹₁ₑ (Z (G) ) Autbr1 (Z (G) ) give rise to different realisations of the charge conjugation modular invariant for G-orbifolds of holomorphic conformal field theories.