Suppose $H$ is a normal subgroup of $G$. If $H$ and $G/H$ are abelian, is it true that $G$ is abelian?
I don't think the answer is YES. This is because, $G/H$ is always abelian as $H$ is normal, so this information is nothing new. $H$ is abelian implies that $H\subset Z(G)$ where $Z(G)$ is the centre of $G$.
But I could not think of a counterexample.