If I understand the question  correctly, you have a map of schemes $G \times X \to X$, and the corresponding map of $k$-points is a group action (meaning that the obvious two maps $G(k) \times G(k) \times X(k) \to X(k)$ coincide), but you are not sure that it is a group action in the category of schemes. In other words, you fear that you may have two maps $G \times G \times X \to X$ which coincide on $k$ points but not as maps of schemes.

This certainly can't happen if $G$ and $X$ are reduced. So, if you are talking about varieties, there is no issue. It's not obvious to me what happens when $G$ is reduced (which is automatic in characteristic zero) but $X$ isn't.