Characteristically complemented characteristic subgroup

Definition
A subgroup of a group is termed a characteristically complemented characteristic subgroup if it is a defining ingredient::characteristic subgroup and it has a permutable complement that is also a characteristic subgroup.

Stronger properties

 * Weaker than::Hall direct factor
 * Weaker than::Sylow direct factor

Weaker properties

 * Stronger than::Left-transitively complemented normal subgroup
 * Stronger than::Characteristic direct factor
 * Stronger than::Characteristically complemented subgroup
 * Stronger than::Complemented characteristic subgroup
 * Stronger than::Complemented normal subgroup
 * Stronger than::Retract
 * Stronger than::Normal subgroup

Metaproperties
Suppose $$H \le K \le G$$ are groups such that $$H$$ is a characteristically complemented characteristic subgroup of $$K$$ and $$K$$ is a characteristically complemented characteristic subgroup of $$G$$. Then $$H$$ is a characteristically complemented characteristic subgroup of $$G$$.