Engel sinks of fixed points in finite groups

For an element g of a group G, an Engel sink is a subset E (g) such that for every x ∈ G all sufficiently long commutators [x,g,g,...,g] belong to E(g). Let q be a prime, let m be a positive integer and A an elementary abelian group of order q^2 acting coprimely on a finite group G. We show that if for each nontrivial element a in A and every element g ∈ C_G(a) the cardinality of the smallest Engel sink E (g) is at most m, then the order of γ∞(G) is bounded in terms of m only. Moreover we prove that if for each a ∈ A {1} and every element g ∈ CG(a), the smallest Engel sink E(g) generates a subgroup of rank at most m, then the rank of γ∞(G) is bounded in terms of m and q only.