Realization of even permutations of even degree by products of four involutions without fixed points

We consider representations of an arbitrary permutation $\pi$ of degree $2n$, $n\geqslant3$, by products of the so-called $(2^n)$-permutations (any cycle of such a permutation has length 2). We show that any even permutation is represented by the product of four $(2^n)$-permutations. Products of three $(2^n)$-permutations cannot represent all even permutations. Any odd permutation is realized (for odd $n$) by a product of five $(2^n)$-permutations.