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

A group word w is said to be strongly concise in a class C of profinite groups if, for any group G in C , either w takes at least continuum values in G or the verbal subgroup w(G) is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups.In the present paper we prove that for any n the n-Engel word [...[x, y], y], . . . y] (where y is repeated n times) is strongly concise in the class of finitely generated profinite groups.

Suppose that a finite group G admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer m, every element of the centralizer C G (A) has a left Engel sink of cardinality at most m (or a right Engel sink of cardinality at most m), then G has a subgroup of (|A|, m)-bounded index which has Fitting height at most 2α(A) + 2, where α(A) is the composition length of A. We also prove that if, for some positive integer r, every element of the centralizer C G (A) has a left Engel sink of rank at most r (or a right Engel sink of rank at most r), then G has a subgroup of (|A|, r)-bounded index which has Fitting height at most 4 α(A) + 4α(A) + 3. Here, a left Engel sink of an element g of a group G is a set E (g) such that for every x ∈ G all sufficiently long commutators [...[[x, g], g], . . . , g] belong to E (g). (Thus, g is a left Engel element precisely when we can choose E (g) = {1}.) A right Engel sink of an element g of a group G is a set R(g) such that for every x ∈ G all sufficiently long commutators [...[[g, x], x], . . . , x] belong to R(g). (Thus, g is a right Engel element precisely when we can choose R(g) = {1}.

A right Engel sink of an element g of a group G is a set Rpgq such that for every x P G all sufficiently long commutators r...rrg, xs, xs, . . . , xs belong to Rpgq. (Thus, g is a right Engel element precisely when we can choose Rpgq " t1u.) We prove that if a profinite group G admits a coprime automorphism ϕ of prime order such that every fixed point of ϕ has a finite right Engel sink, then G has an open locally nilpotent subgroup.A left Engel sink of an element g of a group G is a set E pgq such that for every x P G all sufficiently long commutators r...rrx, gs, gs, . . . , gs belong to E pgq. (Thus, g is a left Engel element precisely when we can choose E pgq " t1u.) We prove that if a profinite group G admits a coprime automorphism ϕ of prime order such that every fixed point of ϕ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.