Commuting probability for the Sylow subgroups of a finite group

For subsets X, Y of a finite group G, let Pr(X, Y) denote the probability that two random elements x ∈ X and y ∈ Y commute. Suppose that G is a finite group in which for any distinct primes p, q ∈ π(G) there is a Sylow p-subgroup P and a Sylow q-subgroup Q of G such that Pr(P, Q) ≥ ϵ. We show that F2(G) has ϵ-bounded index in G. If G is a finite soluble group in which for any prime p ∈ π(G) there is a Sylow p-subgroup P and a Hall p′-subgroup H such that Pr(P, H) ≥ ϵ, then F(G) has ϵ-bounded index in G. Moreover, we establish criteria for nilpotency and solubility of G.