Normality Relation is not Transitive

Theorem
Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $K$ be a normal subgroup of $N$.

Then it is not necessarily the case that $K$ is a normal subgroup of $G$.

Proof
Proof by Counterexample:

Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as:

From Alternating Group on 4 Letters: Normality of Subgroups:
 * $K := \set {e, t, u, v}$ is a normal subgroup of $A_4$
 * $T := \set {e, t}$ is not a normal subgroup of $A_4$.

But by Subgroup of Abelian Group is Normal:
 * $T$ is a normal subgroup of $K$.

Thus we have:
 * $T \lhd K$, $K \lhd A_4$

but:
 * $T \not \lhd A_4$