Finite groups with high commuting probability for Sylow subgroups

Given two subsets $X,Y$ of a finite group $G$, we write $\mathrm{Pr}(X,Y)$ for the probability that random elements $x\in X$ and $y\in Y$ commute. If $X,Y$ are subgroups, we denote by ${\mathrm{Pr}}^{\ast }(X,Y)$ 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 $\mathrm{Pr}(P,Q)\ge \epsilon$. In this paper we handle, among other things, finite groups $G$ with high probabilities ${\mathrm{Pr}}^{\ast }(T,G)$, where $T$ is either a term of the lower central series of $G$ or the generalized Fitting subgroup $F_i^{\ast }(G)$. Our main results show that the structure of such groups is similar, in some precise sense, to that of nilpotent groups.