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 — κ_τ(σ) = τ·σ·τ ; compose left-regular
representation ρ: G → (G/H)! ⊂ E! ({ρ(g)}(xH) = gxH , {ρ(g)}(∞) = ∞ ) with κ_τ: E! → E! ;
for h ∈ H and C ∈ G/H ∪ {∞} ( C = xH (x ∈ G) or C = ∞ ) , calculate {κ_τ·ρ(h)}(C) =

τ({ρ(h)}(τ(C))) = {ρ(h)}(C)