Upward-closed transitively normal subgroup

Symbol-free definition
A subgroup of a group is termed an upward-closed transitively normal subgroup if every subgroup of the whole group containing it is a defining ingredient::transitively normal subgroup of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed an upward-closed transitively normal subgroup if, for every subgroup $$K$$ of $$G$$ containing $$H$$ and every normal subgroup $$L$$ of $$K$$, $$L$$ is normal in $$G$$.