Potentially fully invariant subgroup

Symbol-free definition
A subgroup of a group is termed potentially fully invariant if there is an embedding of the bigger group in some group such that, in that embedding the subgroup becomes fully invariant.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed potentially fully invariant in $$G$$ if there exists a group $$K$$ containing $$G$$ such that $$H$$ is fully invariant in $$K$$.

Formalisms
The property of being potentially fully invariant is obtained by applying the potentially operator to the property of being fully invariant. The potentially operator is an idempotent ascendant monotone operator.

Incomparable properties

 * Characteristic subgroup: See characteristic not implies potentially fully invariant