Let $G$ be a reductive group over a field $k$. Then there is a natural functor from the category of representations of $G$ to the category of representations of $G(k)$.

Under which circumstances can one say that this functor is a) full and b) faithful?

Let $G$ be a (connected) reductive group over a field $k$. Then there is a natural functor from the category of representations of $G$ to the category of representations of $G(k)$.

Under which circumstances can one say that this functor is a) full and b) faithful?

Edited for clarity:

Here we think of algebraic representations of $G$ over the field $k$ (so, for example, morphisms of $G$ to $GL(V)$ for $k$-vector spaces $V$, defined over $k$).

Given such a morphism $\phi : G \rightarrow GL(V)$, we take $k$ points to obtain a $k$-linear representation of the group $G(k)$ on the $k$-vector space $V$ (thus a homomorphism $\phi(k) : G(k) \rightarrow GL(V)(k)$).