Let $G$ be a group, and let $V$ be a finite dimensional $\mathbb Q$-linear representation of $G.$ By extension of scalars we obtain the $\mathbb Q_l$-linear representation $V_l=V\otimes\mathbb Q_l.$
Question: Is the natural morphism ´$$ H^1(G,V)\to\prod_lH^1(G,V_l) $$ injective? Stated in this generality, the answer is possibly negative. But are there some contexts (e.g. G is a $\mathbb Q$-algebraic group and $V$ is an algebraic representation etc.) in which the answer is yes?
And what about other degrees $H^i$?