Transitive normality is not quotient-transitive

Property-theoretic statement
The property of being a transitively normal subgroup is not a quotient-transitive subgroup property.

Statement with symbols
It is possible to have the following situation: $$H \le K \le G$$, $$H$$ is a transitively normal subgroup of a group $$G$$, and $$K/H$$ is a transitively normal subgroup of $$G/H$$, and $$K$$ is not a transitively normal subgroup of $$G$$.

Transitively normal subgroup
We say that $$H$$ is a transitively normal subgroup of $$G$$ if whenever $$K$$ is a normal subgroup of $$H$$, $$K$$ is also a normal subgroup of $$G$$.

Failure of quotient-transitivity for related properties
The same example used here applies to all these related properties:


 * Central factor is not quotient-transitive
 * SCAB is not quotient-transitive
 * Conjugacy-closed normality is not quotient-transitive

Example of the dihedral group
Let $$G$$ be the dihedral group, given by:

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

Define subgroups:

$$H = \langle a^2 \rangle, \qquad K = \langle a^2, x \rangle$$.

$$H$$ is a subgroup of order two, hence it is transitively normal in $$G$$. $$K/H$$ is a subgroup of order two in $$G/H$$, hence it is transitively normal in $$G$$.

However, $$K$$ is not transitively normal in $$G$$, because the subgroup $$\langle x \rangle$$ of $$K$$ is normal in $$K$$ but not in $$G$$.