A general notion of $chi$-transitive groups was introduced by C. Delizia et al. in cite{d}, where $chi$ is a class of groups. In cite{c}, Ciobanu, Fine and Rosenberger studied the relationship among the notions of conjugately separated abelian, commutative transitive and fully residually $chi$-groups. In this article we study the concept of $2$-Engel transitive groups and among other results, its relationship with conjugately separated $2$-Engel and fully residually $chi$-groups are established. We also introduce the notion of $2$-Engelizer of the element $x$ in $G$ and denote the set of all $2$-Engelizers in $G$ by $E^2(G)$. Then we construct the possible values of $|E^2(G)|$.
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机译：C. Delizia等人介绍了$ chi $传递基团的一般概念。在 cite {d}中，其中$ chi $是一组组。 Ciobanu，Fine和Rosenberger在 cite {c}中研究了共轭分离的阿贝尔语，可交换可及和完全残差的$ chi $-组的概念之间的关系。在本文中，我们研究了$ 2 $ -Engel传递基团的概念，以及其他结果，建立了它与共轭分离的$ 2 $ -Engel和完全残差的$ chi $ -groups的关系。我们还介绍了$ G $中元素$ x $的$ 2 $ -Engelizer的概念，并用$ E ^ 2（G）$表示$ G $中所有$ 2 $ -Engelizers的集合。然后我们构造$ | E ^ 2（G）| $的可能值。
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