 We can arrange that the product contains either 
For "k" ≥ 5, adjoin to the permutations "σ"1, "σ"2 of the last example redundant 2-cycles that cancel each other, and we still get two 4-cycles.Now we arrive at a contradiction, because if the class of transpositions is sent via the automorphism "f" to a class of involutions that has "k" > 1, then there exist two transpositions "τ"1, "τ"2 such that "f"("τ"1) "f"("τ"2) has order 6, 7 or 4, but we know that "τ"1"τ"2 has order 2 or 3.S6 has exactly one (class) of outer automorphisms