Solvable normal subgroup but the corresponding quotient group is not solvable

Question

Let $G$ be a group and $N$ be a normal subgroup of $G$. It is a well known fact that: $G$ is solvable iff $N$ and $G/N$ are solvable.

I wonder the following: Could you give an example of a group $G$ and its normal subgroup $N$ such that $N$ is solvable but $G/N$ is not.

Answer 1

Take any solvable group $S$, and non-solvable $G$, and consider $G×S$.

For instance, I was thinking $A_5×C_2$ (but was beat to it in the comments).