Eilenberg on epimorphisms among groups

Group G , subgroup H ⊆ G.   Can that inclusion be an epimorphism?

H of index 1 in G ? — Yes (H = G).
H of index 2 ? — No (H is normal in G).
H of index ≥ 3 ? — Fix two cosets Ha, Hb of H, distinct from each other and from H ,
  and define τ ∈ |G|! — an involution — by giving τ(x), for x ∈ G, as