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

Statement
Suppose $$H \le K \le G$$ are such that $$H$$ is an intermediately normal-to-characteristic subgroup of $$K$$, and $$K$$ is normal in $$G$$. Then, $$H$$ is intermediately subnormal-to-normal in $$K$$.

Weaker facts

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

Converse
It is not in general true that if $$H \le K$$ is a subgroup such that whenever $$K$$ is normal in $$G$$, $$H$$ is intermediately subnormal-to-normal in $$G$$, then H is intermediately normal-to-characteristic in $$K$$. Thus, being intermediately normal-to-characteristic is not the left residual of intermediately subnormal-to-normal by normal.

Facts used

 * 1) uses::Subnormality satisfies intermediate subgroup condition
 * 2) uses::Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
 * 3) uses::Normality satisfies transfer condition: The intersection of a normal subgroup with any subgroup is normal in that subgroup.
 * 4) uses::Characteristic of normal implies normal

Proof
Given: Groups $$H \le K \le G$$ such that $$H$$ is intermediately normal-to-characteristic in $$K$$ and $$K$$ is normal in $$G$$.

To prove: If $$L \le G$$ is such that $$H$$ is subnormal in $$L$$, then $$L$$ is normal in $$L$$.

Proof: Let $$M = K \cap L$$.


 * 1) $$H$$ is characteristic in $$M$$: By fact (1), $$H$$ is subnormal in $$M$$. By fact (2), $$H$$ is normal in $$M$$, and hence characteristic in $$M$$.
 * 2) $$M$$ is normal in $$L$$: Since $$K$$ is normal in $$G$$, fact (3) yields that $$M = K \cap L$$ is normal in $$L$$.
 * 3) $$H$$ is normal in $$L$$: This follows from the previous two steps, and fact (4).