Deleted permutation module in characteristic two ($\{x\in {\mathbb{F}}_2^{2m+2}:\sum x_i=0\}/⟨(1,\dots ,1)⟩$)

For even degree the augmentation hyperplane has an alternating dot product whose radical is the all-ones line. Quotienting by that line gives a nondegenerate symplectic space of dimension two less than the permutation degree. Coordinate permutations preserve the form. In degree at least six their quotient action is faithful, since two distinct weight-two coordinate vectors cannot differ by the all-ones vector. In degree six this gives $S_6\cong S{p}_4(2)$; in degree four the quotient action instead has Klein-four kernel.