  For an element $g$ of a group $G$, an Engel sink is a subset $\mathscr{E}(g)$
such that for every $ x\in 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 $g\in C_{G}(a)$ 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 $a\in A\setminus \{1\}$ and every element $g\in C_{G}(a)$, 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.
