Probabilistically nilpotent groups of class two

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, z] = 1$. We determine structure groups is bounded away from zero: if $d_2(G) \geq \epsilon > 0$, has class-4 nilpotent normal subgroup $H$ $[G : H] $ and $|\gamma_4(H)|$ are both in terms $\epsilon$. also show an infinite group whose commutators have boundedly many conjugates, or indeed satisfies certain more general commutator covering condition, then finite-by-class-3-nilpotent-by-finite.