Left-transitively 2-subnormal subgroup

Definition
A subgroup $$H$$ of a group $$K$$ is termed a left-transitively 2-subnormal subgroup if it satisfies the following equivalent conditions:


 * Whenever $$K$$ is a 2-subnormal subgroup of a group $$G$$, $$H$$ is also a 2-subnormal subgroup of $$G$$.
 * Whenever $$K$$ is a normal subgroup of characteristic subgroup of a group $$G$$, $$H$$ is also a normal subgroup of characteristic subgroup of $$G$$.

Incomparable properties

 * Normal subgroup:
 * Right-transitively 2-subnormal subgroup