The Engel graph of almost simple groups

The Engel graph of a finite group G, denoted Γ(G), is a directed graph encoding the pairs of elements in G satisfying some Engel word. Recent work of Detomi, Lucchini and Nemmi shows that Γ(G) is connected and the problem of understanding the strong connectivity of this graph has been reduced to the case where G is an almost simple group. In this paper, we complete the analysis by determining when Γ(G) is strongly connected in the almost simple setting; the only exceptions involve groups with socle PSL2(q) or 2B2(q).