Engel sinks of fixed points in finite groups

For an element $g$ of a group $G$, an Engel sink is a subset $mathscr{E}(g)$ such that for every $ xin G $ all sufficiently long commutators $ [x,g,g,ldots,g] $ belong to $mathscr{E}(g)$. 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 $gin C_{G}(a)$ the cardinality of the smallest Engel sink $mathscr{E}(g)$ is at most $m$, then the order of $gamma_infty(G)$ is bounded in terms of $m$ only. Moreover we prove that if for each $ain Asetminus {1}$ and every element $gin C_{G}(a)$, the smallest Engel sink $mathscr{E}(g)$ generates a subgroup of rank at most $m$, then the rank of $gamma_infty(G)$ is bounded in terms of $m$ and $q$ only.