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Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as acoprime group of automorphisms on a profinite group $G$. We show that if $A$ isof order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ forany $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Assuming that $A$is of order $q^3$ we prove that if some power of each element in $C_G(a)$ isEngel in $C_G(a)$ for any $a\in A^{\#}$, then $G$ is locally virtuallynilpotent. Some analogues consequences of quantitative nature for finite groupsare also obtained.

Engel-like conditions in fixed points of automorphisms of profinite groups