Finite groups with high commuting probability for Sylow subgroups

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $\epsilon$ with the property that for every pair of distinct primes $p\in\pi(X)$ and $q\in\pi(Y)$ there is a Sylow $p$-subgroup $P$ of $X$ and a Sylow $q$-subgroup $Q$ of $Y$ such that $\Pr(P,Q) \geq \epsilon$. In this paper we handle, among other things, finite groups $G$ with high probabilities $\Pr^*(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^*(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups.