Reflection classes whose cubes cover the alternating group

A reflection class (REC) over a finite set A is a conjugacy class of a reflection (permutation of order ≤ 2) of A. It was known that for no REC X, X² = Alt(n) holds, and that for some RECs X, X⁴ = Alt(n) holds (n ≥ 5). Let i > 0, and let c(θ) denote the number of cycles of θ ε{lunate} S(n). Let X_i = {ψ ∈ S(n): ψ² = 1, ψ has exactly i fixed points}. We prove that θ ε{lunate} X_i³ if and only if: (1) i ≡ n (mod 2); (2) The parity of X_i equals the parity of θ; and (3) i ≤ 1 3(n + 2 c(θ)). As a consequence, {X: X is a REC, X³ = Alt(n)} and {X: X is a REC, X³ = S(n) - Alt(n)} are determined.