The action is fairly straightforward. Any line in $k_E^2$ can be described by an equation of the form $ax+by=c$, where $a,b,c \in k_E$, and $a$ and $b$ are not both zero. You can describe the action of $\operatorname{Gal}(E/F)$ on $k_E$ by taking reduction modulo the maximal ideal. Alternatively, you can look at the action on the (prime-to-residue-characteristic) roots of unity in $E$ living over units in $k_E$. You can then use the action on $k_E$ to get the diagonal action on the coefficients $a,b,c$ in the equation $ax+by=c$.
If you want to restrict your view to lines through the origin, you get an action on pairs $(a,b)$, and the equivalence relation that sends proportional pairs to the same point in $\mathbb{P}^1$ is Galois-stable.