On perfect isometries and isotypies in alternating groups
Paul Fong and Morton E. Harris
Trans. Amer. Math. Soc. 349 (1997), 3469-3516.

Abstract. Perfect isometries and isotypies are constructed for alternating 
groups between blocks with abelian defect groups and the Brauer correspondents 
of these blocks. These perfect isometries and isotypies satisfy additional 
compatibility conditions which imply that an extended Brou\'e conjecture
holds for the principal block of an almost simple group with an abelian 
Sylow p-subgroup and a generalized Fitting subgroup isomorphic to an 
alternating group.