Let define the absolute Frobenius action on cohomology,

Let $ X $ be a scheme over a field $ K$. Fix a separable closure $ K^s $ of $ K $. Let $ G_K = operatorname{Gal} ( K^s / K ) $ be the absolute Galois group of $ K $. Let $ mathcal{F} $ an abelian sheaf on $ X_{ mathrm{et} } $

I try to define properly a left $ G_K $-module structure cohomology group $ H^{j} ( X_{K^{s}} , mathcal{F}_{|K^{s} } ) $ as follows,

If $ sigma in G_K $, then, we can set, for $ xi in H^{j} ( X_{K^{s}} , mathcal{F}_{| X_{K_{s}}} ) $, $$ sigma cdot xi = : ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^* xi in H^{j} ( X_{K^{s}} , ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^{-1} mathcal{F}_{|X_{K^{s}}} ) = H^{j} ( X_{K^{s}} , mathcal{F}_{| X_{K^{s}}} ) $$

So, my question is,

How is defined explicitly $ ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^* xi $ in $ H^{j} ( X_{K^{s}} , ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^{-1} mathcal{F}_{|X_{K^{s}}} ) $ ?

To be more concrete, if $ X = mathrm{Spec} (A) $, and $ A = K (t_1 , dots , t_n ) $ is a finitely generated $ K $ – algebra. then, $ X_{K^{s}} = mathrm{Spec} ( K^s otimes_K A ) = mathrm{Spec} ( K^{s} (t_1 , dots , t_n ) ) $.

So, how to express explicitly $ sigma cdot xi = : ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^* xi in H^{j} ( X_{K^{s}} , ( mathrm{Spec} ( sigma ) times mathrm{id}_X )^{-1} mathcal{F}_{|X_{K^{s}}} ) $ with, $ X_{K^{s}} = mathrm{Spec} ( K^s (t_1 , dots , t_n ) ) $ ?

Other questions:

We defined above the first case of absolute Frobenius action on cohomology, so now, how is defined, the arithmetic, geometric, and relative Frobenius actions on cohomology ?

Thank you.