Central factor implies transitively normal

Verbal statement
Any central factor of a group is a transitively normal subgroup. In other words, any normal subgroup of a central factor is a central factor.

Statement with symbols
Suppose $$K$$ is a central factor of a group $$G$$. Then, $$K$$ is a transitively normal subgroup of $$G$$, i.e., if $$H$$ is a normal subgroup of $$K$$, then $$H$$ is a normal subgroup of $$G$$.

Proof
Given: A group $$G$$, a central factor $$K$$ of $$G$$, a normal subgroup $$H$$ of $$K$$. $$g$$ is an element of $$G$$ and $$h$$ is an element of $$H$$.

To prove: $$ghg^{-1} \in H$$.

Proof: Let $$c_g = x \mapsto gxg^{-1}$$ be the inner automorphism obtained as conjugation by $$g$$.


 * 1) There exists $$k \in K$$ such that the inner automorphism $$c_k$$ by $$k \in K$$, as an inner automorphism of $$K$$, is the restriction to $$K$$ of $$c_g$$. In other words, for any $$x \in K$$, $$gxg^{-1} = kxk^{-1}$$: This follows from the definition of central factor.
 * 2) $$khk^{-1} \in K$$: This follows from the fact that $$K$$ is normal in $$H$$.
 * 3) $$ghg^{-1} \in K$$: This follows immediately from the previous two steps.