Abstract Let q be a prime, n a positive integer and A an elementary abelian group of order q r {q^{r}} with r ≥ 2 {r\geq 2} acting on a finite q ′ {q^{\prime}} -group G. We show that if all elements in γ r - 1 ( C G ( a ) ) {\gamma_{r-1}(C_{G}(a))} are n-Engel in G for any a ∈ A # {a\in A^{\#}} , then γ r - 1 ( G ) {\gamma_{r-1}(G)} is k-Engel for some { n , q , r } {\{n,q,r\}} -bounded number k, and if, for some integer d such that 2 d ≤ r - 1 {2^{d}\leq r-1} , all elements in the dth derived group of C G ( a ) {C_{G}(a)} are n-Engel in G for any a ∈ A # {a\in A^{\#}} , then the dth derived group G ( d ) {G^{(d)}} is k-Engel for some { n , q , r } {\{n,q,r\}} -bounded number k. Assuming r ≥ 3 {r\geq 3} , we prove that if all elements in γ r - 2 ( C G ( a ) ) {\gamma_{r-2}(C_{G}(a))} are n-Engel in C G ( a ) {C_{G}(a)} for any a ∈ A # {a\in A^{\#}} , then γ r - 2 ( G ) {\gamma_{r-2}(G)} is k-Engel for some { n , q , r } {\{n,q,r\}} -bounded number k, and if, for some integer d such that 2 d ≤ r - 2 {2^{d}\leq r-2} , all elements in the dth derived group of C G ( a ) {C_{G}(a)} are n-Engel in C G ( a ) {C_{G}(a)} for any a ∈ A # , {a\in A^{\#},} then the dth derived group G ( d ) {G^{(d)}} is k-Engel for some { n , q , r } {\{n,q,r\}} -bounded number k. Analogous (non-quantitative) results for profinite groups are also obtained.