Left-transitively fixed-depth subnormal subgroup

Definition
A subgroup $$H$$ of a group $$K$$ is termed left-transitively fixed-depth subnormal in $$K$$ if there exists a natural number $$k \ge 1$$ such that $$H$$ is left-transitively $$k$$-subnormal in $$K$$. In other words, whenever $$K$$ is a $$k$$-subnormal subgroup of a group $$G$$, $$H$$ is also $$k$$-subnormal in $$G$$.

Note that any subgroup that is left-transitively $$k$$-subnormal is also left-transitively $$l$$-subnormal for $$l \ge k$$.

Metaproperties
If $$H \le K \le G$$ are such that $$H$$ is left-transitively $$k$$-subnormal in $$K$$ and $$K$$ is left-transitively $$l$$-subnormal in $$G$$, then $$H$$ is left-transitively $$\max \{ k,l \}$$-subnormal in $$G$$.

An intersection of finitely many such subgroups again has the property. In particular, the intersection of a left-transitively $$k$$-subnormal subgroup and a left-transitively $$l$$-subnormal subgroup is left-transitively $$\max\{k,l \}$$-subnormal.

A join of finitely many such subgroups again has the property. In particular, the join of a left-transitively $$k$$-subnormal subgroup and a left-transitively $$l$$-subnormal subgroup is left-transitively $$\max\{k,l \}$$-subnormal.