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

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.