Are normal subgroups transitive?

Using some suggestions from the other commenters:

The alternating group, $A_4$, has the set $H=\{I,(12)(34),(13)(24),(14)(23)\}\cong V_4$ as a subgroup. If $f\in S_4\supseteq A_4$ is a permutation, then $f^{-1}[(12)(34)]f$ has the effect of swapping $f(1)$ with $f(2)$ and $f(3)$ with $f(4)$. One of these is $1$, and depending on which it is paired with, the conjugated element may be any of $H-\{I\}$, since the other two are also swapped. Thus $H\lhd S_4$ is normal, so $H\lhd A_4$ as well. Similarly, $H$ has three nontrivial subgroups, and taking $K=\{I,(12)(34)\}\cong C_2$, this is normal because $V_4$ is abelian. But $K\not\lhd A_4$, since $$[(123)][(12)(34)][(132)]=(13)(24)\in H-K.$$

Moreover, this is a minimal counterexample, since $|A_4|=12=2\cdot 2\cdot 3$ is the next smallest number which factors into three integers, which is required for $K\lhd H\lhd G$ but $\{I\}\subset K\subset H\subset G$ so that $[G\,:\,H]>1$, $[H\,:\,K]>1$, $|K|>1$ and $$|G|=[G\,:\,H]\cdot[H\,:\,K]\cdot|K|.$$ The smallest integer satisfying this requirement is 8, but the only non-abelian groups with $|G|=8$ are the dihedral group $D_4$ and the quaternion group $Q_8$, and neither of these have counterexamples. (Note that if $G$ is abelian, then all subgroups are normal.) Thus $A_4$ is a minimal counterexample. (Edit: Oops, $D_4$ has a counterexample, as mentioned in the comments: $\langle s\rangle\lhd\langle r^2,s\rangle\lhd \langle r,s\rangle=D_4$, but $\langle s\rangle\not\lhd D_4$.)

However, if $H\lhd G$ and $K$ is a characteristic subgroup of $H$, then $K$ is normal in $G$. This is because the group action $f$ defines an automorphism on $G$, $\varphi(g)=f^{-1}gf$, and because $H$ is normal, $\varphi(H)=H$ so that $\varphi|_H$ is an automorphism on $H$. Thus $\varphi(K)=K$ since $K$ is characteristic on $H$ and so $\{f^{-1}kf\mid k\in K\}=K\Rightarrow K\lhd G$.

Look at $S_4$ and its following subgroups $A = \langle (12)(34) \rangle$ and $B=\{(12)(34),(13)(42),(23)(41),e \}$. Try to show that $A$ is normal in $B$ and $B$ is normal in $S_4$ but $A$ is not normal in $S_4$.

We need a non-abelian group, since all subgroups of abelian groups are normal. One small candidate is $D_8$, the symmetries of a square, here in a little more detail about how we might go about finding examples:

Consider all the subgroups in $D_8$. It's useful to visualize the subgroups as a lattice:

(Picture of Dummit and Foote I found on the web)

Now we try to pick an $H$. For all the subgroups on the third row from the top, their only proper subgroup is the trivial subgroup, which is trivially normal to $G$, so it doesn't make sense to use any of the subgroups on the third row for $H$.

Our only options for $H$ now are the second row: $\langle s, r^2 \rangle$, $\langle r \rangle$, and $\langle rs, r^2 \rangle$. We observe that if $H = \langle r \rangle$, the proper subgroups $\langle r^2 \rangle$ and $1$ are both normal to $D_8$, so that case is excluded. Our candidates are $\langle s, r^2 \rangle$ and $\langle rs, r^2 \rangle$.

Take $H = \langle s, r^2 \rangle$ and $K = \langle s \rangle$. It's easy to verify that $\langle s \rangle$ is not normal to $D_8$. All that's left is to show $K \lhd H$ and $H \lhd G$.

This is not difficult if we remember that any element in $D_8$ can be written as $r^i s^j$ with $0 \le i \le 3$ and $j = 0, 1$. Also we have the identity $rs = sr^{-1}$ which can repeated as $r^k s = s r^{-k}$. So for $g \in D_8$, we want to prove or disprove $r^i s^j n s^j r^{-i} \in N$ to show $N \lhd G$.