Prove that if there exists an ascending chain of subfields of E such that [E_(i):E_(i-1)]=2 for all i if and only if [E:K] is a power of 2.

Question

Let $F/K$ be a Galois extension, and let $E$ be an intermediate field; $E$ is said to be a $2$-tower over $K$ if there exists an ascending chain of subfields between $K$ and $E$ such that $[E_i:E_{i-1}]=2$ for all $i$. Suppose that $Aut(E:K$) is abelian. Prove that $E$ is a $2$-tower over $K$ if and only if $[E:K]$ is a power of $2$.