Extensions of Rickard equivalences

Broué conjectures that a block of a finite group G with abelian defect group P   is Rickard equivalent to its Brauer correspondent in the normalizer NG(P)NG(P). We find a character-theoretic criterion on extensions of Rickard equivalences between blocks. As an application, we prove that the conjecture holds for the principal block of a finite group with abelian defect group and cyclic hyperfocal subgroup.