Let us fix a base field $ k $, and denote by $ \mathcal{P} (k) $ the category of smooth projective schemes on $ k $, sometimes called, $ k $-smooth projective varieties in the following.
For all $ X \in \mathcal{P} (k) $, denote by $ \mathcal{Z}^* (X) $ the graduated group of algebraic cycles over $ X $.
For any commutative ring $ F $, the elements of, $$ \mathcal{Z}^r (X)_F: = \mathcal{Z}^r (X) \otimes_{\mathbb{Z}} F $$ are called, algebraic cycles of codimension $ r $, with coefficients in $ F $.
Let $ \sim $ be an adequate equivalence relation over the algebraic cycles of, $ \mathcal{Z}^r (X)_F $, such that, $$ \mathcal{Z}_{\sim}^* (X)_F: = \mathcal{Z}^* (X)_F / \sim. $$ Let $ k '/ k $ be an extension of field. Let us start from an adequate equivalence $ \sim $ for the algebraic cycles on the objects of $ \mathcal{P} (k ') $. According to the course of Yves André, Introduction to motives's theory, $ \sim $ induces, by restriction, an adequate equivalence (again denoted $ \sim $) for the algebraic cycles $ \alpha $ on the objects of $ \mathcal{P} (k) \ : \ \alpha \sim 0 \ \Longleftrightarrow \ \alpha_{k '} \sim 0 $.
Why does this adequate equivalence induce an injective canonical homomorphism, $$ \mathcal{Z}_{\sim}^* (X)_F \to \mathcal{Z}_{\sim}^* (X_{k '})_F ? $$
For $ k '= \overline{k} $, how does the Galois group $ Gal (\overline{k} / k) $ naturally act on $ \mathcal{Z}_{\sim }^* (X_{\overline{k}})_F $?
How does the previous homomorphism induce another morphism of the form, $$ \mathcal{Z}_{\sim}^* (X)_F \to \mathcal{Z}_{\sim}^* (X_{\overline{k}})_F^{Gal (\overline{k} / k)} ? $$
Why is this homomorphism bijectif if $ F $ is a $ \mathbb{Q} $ -algebra, but not surjective in general, if $ F = \mathbb{Z} $?
Thanks in advance for your help.