Nilpotent residual of fixed points

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