$N \lhd G, G/N$ are Nilpotent $ \not \Rightarrow G$ is Nilpotent

A classic example is that $S_3$ is not nilpotent, but $A_3\cong\Bbb{Z}/3\Bbb{Z}$ and $S_3/A_3 \cong \Bbb{Z}/2\Bbb{Z}$ are both nilpotent, and $A_3\lhd S_3.$ Thus, $S_3$ is an example of such a group.