Finite groups, commuting probability, and coprime automorphisms

Given two subgroups $H,K$ of a finite group $G$, the probability that a pair of random elements from $H$ and $K$ commutes is denoted by $Pr(H,K)$. Suppose that a finite group $G$ admits a group of coprime automorphisms $A$ and let $\epsilon>0$. We show that, if for any distinct primes $p,q\in\pi(G)$ there is an $A$-invariant Sylow $p$-subgroup $P$ and an $A$-invariant Sylow $q$-subgroup $Q$ of $G$ for which $Pr([P,A],[Q,A])\ge\epsilon$, then $F_2([G,A])$ has $\epsilon$-bounded index in $[G,A]$ (Theorem 1.2). Here $F_2(K)$ stands for the second term of the upper Fitting seris of a group $K$. We also show that, if $G=[G,A]$ and for any prime $p$ dividing the order of $G$ there is an $A$-invariant Sylow $p$-subgroup $P$ such that $\Pr([P,A], [P,A]^x)\geq\epsilon$ for all $x\in G$, then $G$ is bounded-by-abelian-by-bounded (Theorem 1.4).