Finite Groups with Specific Number of 2-Engelizers

In 2016, the second and third authors introduced the notion of 2-Engelizer of the element x in a given group G and denoted the set of all 2-Engelizers in G by E 2 (G).They also constructed the possible values of |E 2 (G)| [Bull.Korean Math.Soc., 53(3), (2016), 657-665].In the present paper, we classify all non 2-Engel finite groups G, when |E 2 (G)| = 4, 5.