Let $K\supset \mathbb Q$ a finite extension such that $G:=\operatorname{Gal}(K|\mathbb Q)=\mathbb F_{2^n}$, and let $w$ be an absolute value of $K$ extending a non-archimedean absolute value of $\mathbb Q$.
Consider now the decomposition group of $w$:
$$ G_w:=\{\sigma\in G\colon w\circ\sigma=w\} $$
Is it true that in this case $|G_w|\le 2$? One should show that $G_w$ is cyclic, but I don't know how to proceed.