Let $k$ be a perfect field of characteristic $p$. Let also $G$ be a (affine) group scheme over $k$, and $V$ be a representation of $G$.
Given a scheme $X$ over $k$, one can define its Frobenius twist $X^{(1)}$. (See, for instance, here.) This construction is functorial (and symmetric monoidal): In particular, $G^{(1)}$ is also a group scheme, and $V^{(1)}$ is naturally a representation of $G^{(1)}$.
Moreover, there is a canonical map $X\rightarrow X^{(1)}$, so that we may in fact view $V^{(1)}$ as a representation of $G$, via $G\rightarrow G^{(1)}$
The assignment $V\mapsto V^{(1)}$ defines a functor $F$ from the category $\mathbf{Rep}(G)$ of representations of $G$ to itself.
Is the functor $F:\mathbf{Rep}(G)\rightarrow \mathbf{Rep}(G)$ always fully faithful? And if so, what would be a reference for this fact?