Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Question

Have not been able to think of a examples with the following properties:

1. Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of $G$.

2. Example of an infinite group $G$ with property that $G/N$ is finite for every non-trivial normal subgroup $N$ of $G$.

Also, please explain why.

Any help would be greatly appreciated.

Answer 1

Hint: (1) $Q_8, S_3.$ (2) $\mathbb Z.$