Transitively normal not implies central factor

Statement
It is possible to have a group $$G$$ and a transitively normal subgroup $$K$$ of $$G$$ (i.e., any normal subgroup $$H$$ of $$K$$ is normal in $$G$$) that is not a central factor of $$G$$.

Example of the dihedral group
Consider the dihedral group of order eight:

$$G := \langle a,x \mid a^4 = x^2 = e, xax = a^{-1} \rangle$$.

Suppose $$K$$ is the cyclic subgroup of order four:

$$K := \langle a \rangle$$.

Then:


 * $$K$$ is a transitively normal subgroup of $$G$$: $$K$$ is normal (it has index two in $$G$$), and its proper subgroups (the trivial subgroup, and the subgroup $$\langle a^2 \rangle$$) are all normal in $$G$$ as well.
 * $$K$$ is not a central factor of $$G$$: Conjugation by $$x$$ gives an automorphism of $$K$$ that sends $$a$$ to $$a^{-1}$$, and is hence not an inner automorphism of $$K$$.