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 ∪ {∞} ,
have τ ∈ E! interchange H with ∞ but leave the rest of G/H alone. Then do as before:
  Write κ_τ: E! → E! for conjugation by the involution τ —