Transfer-closed characteristic subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a transfer-closed characteristic subgroup if, for any subgroup $$K \le G$$, $$H \cap K$$ is a defining ingredient::characteristic subgroup of $$K$$.

Stronger properties

 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Variety-containing subgroup
 * Weaker than::Subisomorph-containing subgroup

Weaker properties

 * Stronger than::Intermediately characteristic subgroup
 * Stronger than::Characteristic subgroup

Metaproperties
Suppose $$H \le K \le G$$ are groups such that $$K$$ is a transfer-closed characteristic subgroup of $$G$$ and $$H$$ is a transfer-closed characteristic subgroup of $$K$$. Then, $$H$$ is a transfer-closed characteristic subgroup of $$G$$.