Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent

Statement
In a nilpotent group, any intermediately normal-to-characteristic subgroup is an intermediately characteristic subgroup.

Intermediately normal-to-characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed intermediately normal-to-characteristic in $$G$$ if whenever $$H \le K \le G$$, with $$H$$ a normal subgroup of $$K$$, $$H$$ is a characteristic subgroup of $$K$$.

Intermediately characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed intermediately characteristic in $$G$$ if whenever $$H \le K \le G$$, $$H$$ is a characteristic subgroup of $$K$$.

Intermediately subnormal-to-normal subgroup
A subgroup $$H$$ of a group $$G$$ is termed intermediately subnormal-to-normal in $$G$$ if whenever $$H \le K \le G$$, with $$H$$ a subnormal subgroup of $$K$$, $$H$$ is a normal subgroup of $$K$$.

Corollaries

 * Intermediately automorph-conjugate implies intermediately characteristic in nilpotent

Facts used

 * 1) uses::Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
 * 2) uses::Nilpotence is subgroup-closed
 * 3) uses::Nilpotent implies every subgroup is subnormal

Proof
Given: A nilpotent group $$G$$, an intermediately normal-to-characteristic subgroup $$H$$ of $$G$$.

To prove: Whenever $$K$$ is an intermediate subgroup, i.e. $$H \le K \le G$$, $$H$$ is characteristic in $$K$$.

Proof: By fact (2), $$K$$ is nilpotent and by fact (3), $$H$$ is a subnormal subgroup of $$K$$.

By fact (1), $$H$$ is intermediately subnormal-to-normal in $$G$$, and combinig this with the fact that $$H$$ is subnormal in $$K$$ yields that $$H$$ is normal in $$K$$. Combining this with the given fact that $$H$$ is intermediately normal-to-characteristic in $$G$$ yields that $$H$$ is characteristic in $$K$$, completing the proof.