Let consider the one point field scheme $Spec(K)$ and denote by $G:=\text{Gal}(\overline{K}/K)$ the corresponding Galois group.
We consider by $\mathbf{Sh}\big(\text{Spec }k)_{\text{et}}$ the category of etale sheaves on $Spec(k)$. Then there is a wll known theorem that the exist a cetegory equivalence
$$ \mathbf{Sh}\big(\text{Spec }K)_{\text{et}} \cong\{\text{discrete abelian group with continuous }G_K\text{-action}\} $$
given explicitely by
$$\mathcal F \mapsto \varinjlim_{K'/K\text{ finite separable extension}} \mathcal F(\text{Spec }K') $$
with inverse $$\big(\text{Spec }k'\mapsto M^{\text{Gal}(\overline K/K')}\big) \leftarrow M $$
My question is how $G_K$ acts explicitely on $\varinjlim_{K'/K\text{ finite separable extension}} \mathcal F(\text{Spec }K')$?
So in other words if we take a separable field extension $K \subset K'$ what does $G_K$ with $\mathcal F(\text{Spec }K')$?