As KConrad points out, you perhaps mean to say that $F$ is a *finite* extension of $\mathbf{Q}_2$ or of $\mathbf{F}_2((x))$, and that the quadratic extesnions $E|F$ is *separable* (and hence galoisian) in the second case.

With this interpretation of the question, $N_{E|F}(E^*)$ is a closed subgroup of index $2$ in $F^*$, and every closed subgroup of index $2$ in $F^*$ is of this form.  In particular, the ramification index $e_{E|F}$ does *not* determine the subgroup in question.

For more on this, see the relevant chapter in Serre's *Corps locaux* (=Local fields) or the book by Fesenko and Vostokov, among many other places.