Probabilistically nilpotent groups of class two

For G a finite group, let d(2)(G) denote the proportion of triples (x, y, z) is an element of G(3) such that [x, y, z] = 1. We determine the structure of finite groups G such that d(2)(G) is bounded away from zero: if d(2)(G) >= E > 0, G has a class-4 nilpotent normal subgroup H such that [G : H] and |gamma 4(H)| are both bounded in terms of E. We also show that if G is an infinite group whose commutators have boundedly many conjugates, or indeed if G satisfies a certain more general commutator covering condition, then G is finite-by-class-3-nilpotent-by-finite.