Characteristic central factor of WNSCDIN implies WNSCDIN

Statement
Suppose $$H \le K \le G$$ are groups. Suppose $$H$$ is a characteristic central factor of $$K$$: in other words, $$H$$ is both a fact about::characteristic subgroup and a fact about::central factor of $$K$$. Suppose $$K$$ is a WNSCDIN-subgroup of $$G$$. Then, $$H$$ is also a WNSCDIN-subgroup of $$G$$.

In other words, if $$H$$ is a characteristic central factor of $$K$$, then $$H$$ is a left-transitively WNSCDIN-subgroup of $$K$$.

Facts used

 * 1) uses::Characteristic of normal implies normal

Proof
Given: A group $$G$$, a WNSCDIN-subgroup $$K$$ of $$G$$. A characteristic central factor $$H$$ of $$K$$.

To prove: $$H$$ is a WNSCDIN-subgroup of $$G$$. In other words, if $$A$$ and $$B$$ are normal subsets of $$H$$ that are conjugate in $$G$$, then $$A$$ and $$B$$ are conjugate in $$N_G(H)$$.

Proof:


 * 1) $$A,B$$ are normal subsets of $$K$$: Any inner automorphism of $$K$$ restricts to an inner automorphism of $$H$$. In particular, since $$A$$ is invariant under all inner automorphisms of $$H$$, it is invariant under all inner automorphisms of $$H$$. Thus, $$A$$ (and similarly $$B$$), are normal subsets of $$K$$.
 * 2) $$A,B$$ are conjugate in $$N_G(K)$$: This follows from the previous step, and the fact that $$K$$ is WNSCDIN in $$G$$.
 * 3) $$N_G(K) \le N_G(H)$$: $$H$$ is characteristic in $$K$$ and $$K$$ is normal in $$N_G(K)$$. Thus, by fact (1), $$H$$ is normal in $$N_G(K)$$. Thus, $$N_G(K) \le N_G(H)$$.
 * 4) $$A,B$$ are conjugate in $$N_G(H)$$: This follows from the previous two steps.