  Let $q$ be a prime and $A$ an elementary abelian group of order at least
$q^3$ acting by automorphisms on a finite $q'$-group $G$. It is proved that if
$|\gamma_{\infty}(C_{G}(a))|\leq m$ for any $a\in A^{\#}$, then the order of
$\gamma_{\infty}(G)$ 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.
