On the direct products of skew-morphisms

A skew-morphism φ of a finite group G is a permutation on G fixing the identity element of G and for which there is an integer-valued function π on G such that φ(gh) = φ(g)φπ(g)(h) for all g, h ∈ G. For two permutations α : A → A and β : B → B on the sets A and B, their direct product α × β is the permutation on the Cartesian product A × B given by (α × β)(a, b) = (α(a), β(b)) for all (a, b) ∈ A × B. In this paper, necessary and sufficient conditions for a direct product of two skew-morphisms to still be a skew-morphism are given.