Group with unique largest Abelian characteristic subgroup

Definition
A group $$G$$ is said to have unique largest Abelian characteristic subgroup if there exists an Abelian characteristic subgroup $$K \le G$$ such that for every Abelian characteristic subgroup $$H \le G$$, $$H$$ is contained in $$K$$.

Stronger properties

 * Weaker than::Abelian group
 * Weaker than::Simple group
 * Weaker than::Characteristically simple group
 * Weaker than::Characteristic-comparable group