Let $G$ be a finite solvable group and $N$ be a normal subgroup of it which is an elementary abelian 2-group. Suppose that $G/N\cong \mathbb{Z}_2\times \mathbb{Z}_2$ and $|C_G(x)|=16$ for any $x\in G-N$. What can we say about $G$? Is it abelian? Is it elementary?

We know that since $G$ is a 2-group, it is nilpotent.