Left residual of 2-subnormal by normal is normal of characteristic

Statement
Suppose $$H$$ is a subgroup of a group $$G$$. The following are equivalent:


 * 1) $$H$$ is a fact about::normal subgroup of characteristic subgroup of $$G$$: in other words, there is a characteristic subgroup $$K$$ of $$G$$ such that $$H$$ is a normal subgroup of $$K$$.
 * 2) Whenever $$G$$ is a fact about::normal subgroup of some group $$L$$, $$H$$ is a fact about::2-subnormal subgroup of $$L$$.

Related facts

 * Left transiter of normal is characteristic
 * Equivalence of definitions of left-transitively 2-subnormal