  The signed permutation modules are a simultaneous generalization of the
ordinary permutation modules and the twisted permutation modules of the
symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new
invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group
$G$ which attempts to quantify how close a module is to being projective. In
this paper, we determine this invariant for all the signed permutation modules
of the symmetric group using tools from representation theory and
combinatorics.
