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O. Brunat and J. Gramain recently proved that any two blocks of double coversof symmetric groups are Brou\'{e} perfectly isometric provided they have thesame weight and sign. They also proved a corresponding statement for doublecovers of alternating groups and Brou\'{e} perfect isometries between doublecovers of symmetric and alternating groups when the blocks have opposite signs.Using both the results and methods of O. Brunat and J. Gramain in this paper weprove that when the weight of a block of a double cover of a symmetric oralternating group is less than $p$ then the block is Brou\'{e} perfectlyisometric to its Brauer correspondent. This means that Brou\'{e}'s perfectisometry conjecture holds for the double covers of the symmetric andalternating groups. We also explicitly construct the characters of these Brauercorrespondents which may be of independent interest to the reader.

Brou\'e's perfect isometry conjecture holds for the double covers of the symmetric and alternating groups