  The main result of the paper is the following theorem. Let $q$ be a prime and
$A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely
on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each
$a\in A^{\#}$. Then the group $G$ is locally nilpotent.
