Eilenberg, at NSF summer institute, Bowdoin College:

  Why treat index 2 as special case?
  Why mix left regular representation of G with right cosets of H?
  If you need room for a good involutory permutation τ, just make room —
add another point ∞ to the space G/H of left cosets gH of H , forming E = G/H ∪ {∞} ,