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An Engel sink of an element $g$ of a group $G$ is a set ${\mathscr E}(g)$such that for every $x\in G$ all sufficiently long commutators$[...[[x,g],g],\dots ,g]$ belong to ${\mathscr E}(g)$. (Thus, $g$ is an Engelelement precisely when we can choose ${\mathscr E}(g)=\{ 1\}$.) It is provedthat if a profinite group $G$ admits an elementary abelian group ofautomorphisms $A$ of coprime order $q^2$ for a prime $q$ such that for each$a\in A\setminus\{1\}$ every element of the centralizer $C_G(a)$ has acountable (or finite) Engel sink, then $G$ has a finite normal subgroup $N$such that $G/N$ is locally nilpotent.

On profinite groups with automorphisms whose fixed points have countable Engel sinks