If $E/F$ is a Galois extension with abelian Galois group, then $E$ is a tower of quadratic extension iff $[E:F]$ is a power of $2$

Question

I am trying to understand the proof of:

If $E/F$ is a Galois extension with abelian Galois group, then $E$ is a tower of quadratic extension iff $[E:F]$ is a power of $2$.

It is the proof above. But I cannot understand why we must use quotient here. I think even the subgroup of order $2$ is not normal, we still have the $[E^{<\sigma>}:F]=2^{j-1}$. So why do we need the condition that $G$ is abelian and we need to use the quotient?