Left-transitively 2-subnormal implies normal of characteristic

Statement with symbols
Suppose $$H$$ is a subgroup of a group $$K$$ such that whenever $$K$$ is a 2-subnormal subgroup of a group $$G$$, so is $$H$$. Then, $$H$$ is a normal subgroup of characteristic subgroup of $$K$$; in other words, $$H$$ is a normal subgroup of a characteristic subgroup of $$K$$.

Related facts

 * Characteristic of normal implies normal
 * Left transiter of normal is characteristic
 * Left residual of 2-subnormal by normal is normal of characteristic
 * Characteristic implies left-transitively 2-subnormal

Facts used

 * 1) uses::Left residual of 2-subnormal by normal is normal of characteristic

Proof
Given: $$H$$ is a subgroup of $$K$$ such that whenever $$K$$ is a 2-subnormal subgroup of a group $$G$$, so is $$H$$.

To prove: $$H$$ is a normal subgroup of a characteristic subgroup of $$K$$.

Proof: Fact (1) states that if $$H$$ is a subgroup of $$K$$ such that whenever $$K$$ is a normal subgroup of some bigger group $$G$$, $$H$$ is 2-subnormal in $$G$$, then $$H$$ must be a normal subgroup of a characteristic subgroup of $$K$$.

What we're given is that whenever $$K$$ is 2-subnormal in $$G$$ so is $$H$$. Since normal subgroups are 2-subnormal, fact (1) applies to yield that $$H$$ is a normal subgroup of a characteristic subgroup of $$K$$.