Nilpotent residual and Fitting subgroup of fixed points in finite groups

Let $q$ be a prime and $A$ a finite $q$-group of exponent $q$ acting by automorphisms on a finite $q^{\mathrm{\prime }}$-group $G$. Assume that $A$ has order at least $q^3$. We show that if ${\gamma }_{\mathrm{\infty }}(C_G(a))$ has order at most $m$ for any $a\in A^{\mathrm{\#}}$, then the order of ${\gamma }_{\mathrm{\infty }}(G)$ is bounded solely in terms of $m$. If the Fitting subgroup of $C_G(a)$ has index at most $m$ for any $a\in A^{\mathrm{\#}}$, then the second Fitting subgroup of $G$ has index bounded solely in terms of $m$.