On groups with automorphisms whose fixed points are Engel

We study finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let q be a prime and A an elementary abelian q-group of order q(2) acting coprimely on a profinite group G. Assume that all elements in C-G(a) are Engel in G for each a is an element of A(#). Then, G is locally nilpotent. Let q be a prime, n a positive integer and A an elementary abelian group of order q(3) acting coprimely on a finite group G. Assume that for each a is an element of A(#) every element of C-G(a) is n-Engel in C-G(a). Then, the group G is k-Engel for some {n, q}-bounded number