Subgroup having a pronormalizer

Definition
A subgroup $$H$$ of a group $$G$$ is termed a subgroup having a pronormalizer if $$H$$ has a defining ingredient::pronormalizer in $$G$$: a subgroup $$K$$ of $$G$$ containing $$H$$ such that $$H$$ is pronormal in $$K$$, and if $$H \le L \le G$$ are such that $$H$$ is pronormal in $$L$$, then $$L \le K$$.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::Subnormal subgroup
 * Weaker than::Pronormal subgroup