Transitively normal subgroup of normal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a transitively normal subgroup of normal subgroup if there exists a subgroup $$K$$ of $$G$$ containing $$H$$ such that $$K$$ is a normal subgroup of $$G$$ and $$H$$ is a transitively normal subgroup of $$K$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Direct factor of normal subgroup
 * Weaker than::Base of a wreath product
 * Weaker than::Base of a wreath product with diagonal action

Weaker properties

 * Stronger than::2-subnormal subgroup