Let $q$ be a prime and let $A$ be an elementary abelian group of order at least $q^{3}$ acting by automorphisms on a finite $q^{\prime }$-group $G$. We prove that if $|\unicode[STIX]{x1D6FE}_{\infty }(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of $\unicode[STIX]{x1D6FE}_{\infty }(G)$ is $m$-bounded. If $F(C_{G}(a))$ has index at most $m$ in $C_{G}(a)$ for any $a\in A^{\#}$, then the index of $F_{2}(G)$ is $m$-bounded.

The last term of the lower central series of a finite group $G$ is called the nilpotent residual. It is usually denoted by $\unicode[STIX]{x1D6FE}_{\infty }(G)$. The lower Fitting series of $G$ is defined by $D_{0}(G)=G$ and $D_{i+1}(G)=\unicode[STIX]{x1D6FE}_{\infty }(D_{i}(G))$ for $i=0,1,2,\ldots \,$. These subgroups are generated by so-called coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ and $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ in elements of $G$. More precisely, the set of coprime commutators $\unicode[STIX]{x1D6FE}_{k}^{\ast }$ generates $\unicode[STIX]{x1D6FE}_{\infty }(G)$ whenever $k\geq 2$ while the set $\unicode[STIX]{x1D6FF}_{k}^{\ast }$ generates $D_{k}(G)$ for $k\geq 0$. The main result of this article is the following theorem: let $m$ be a positive integer and $G$ a finite group. Let $X\subset G$ be either the set of all $\unicode[STIX]{x1D6FE}_{k}^{\ast }$-commutators for some fixed $k\geq 2$ or the set of all $\unicode[STIX]{x1D6FF}_{k}^{\ast }$-commutators for some fixed $k\geq 1$. Suppose that the size of $a^{X}$ is at most $m$ for any $a\in G$. Then the order of $\langle X\rangle$ is $(k,m)$-bounded.