  O. Brunat and J. Gramain recently proved that any two blocks of double covers
of symmetric groups are Brou\'{e} perfectly isometric provided they have the
same weight and sign. They also proved a corresponding statement for double
covers of alternating groups and Brou\'{e} perfect isometries between double
covers 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 we
prove that when the weight of a block of a double cover of a symmetric or
alternating group is less than $p$ then the block is Brou\'{e} perfectly
isometric to its Brauer correspondent. This means that Brou\'{e}'s perfect
isometry conjecture holds for the double covers of the symmetric and
alternating groups. We also explicitly construct the characters of these Brauer
correspondents which may be of independent interest to the reader.
