Intermediately normal-to-characteristic implies intermediately subnormal-to-normal

Property-theoretic statement
The subgroup property of being intermediately normal-to-characteristic implies the subgroup property of being intermediately subnormal-to-normal.

Intermediately subnormal-to-normal subgroup
A subgroup $$H$$ of a group $$G$$ is termed intermediately subnormal-to-normal if whenever $$K$$ is a subgroup containing $$H$$ such that $$H$$ is 2-subnormal in $$K$$, then $$H$$ is normal in $$K$$.

Intermediately normal-to-characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed intermediately normal-to-characteristic if whenever $$K$$ is a subgroup containing $$H$$ such that $$H$$ is normal in $$K$$, $$H$$ is characteristic in $$K$$.

Facts used

 * 1) uses::Characteristic of normal implies normal

Proof
Given: A subgroup $$H$$ of a group $$G$$, such that $$H$$ is intermediately normal-to-characteristic in $$G$$.

To prove: For any subgroup $$K$$ of $$G$$ containing $$H$$, such that $$H$$ is 2-subnormal in $$K$$, $$H$$ is normal in $$K$$.

Proof: Since $$H$$ is 2-subnormal in $$K$$, there exists a subgroup $$L$$ such that $$H \triangleleft L \triangleleft K$$.

Now, $$L$$ is a subgroup of $$G$$, and $$H$$ is normal in $$L$$. Since $$H$$ is intermediately normal-to-characteristic in $$G$$, $$H$$ is characteristic in $$L$$.

Thus, $$H$$ is characteristic in $$L$$, that in turn is normal in $$K$$. By fact (1), $$H$$ is normal in $$K$$, completing the proof.