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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.

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.

Based on comments below, and on reading the motivating question, I didn't understand right. The question is, given an action $G(k) \times X(k) \to X(k)$, so that $g \times X(k) \to X(k)$ is algebraic for every $k \in G(k)$, can we conclude that it comes from an algebraic map $G \times X \to X$. But I don't think there is any good way to force this. For example, suppose that $k$ has a nontrivial automorphism $\sigma$ and $G$ is defined over the fixed field of $\sigma$. (Think of complex conjugation.) Then $\sigma$ induces an automorphism of $G(k)$ as an abstract group. Take any algebraic action $G \times X \to X$ and compose with the automorphism of $k$ to get a very nonalgebraic action of $G(k)$ on $X(k)$.

If I understand the question in the comment 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 $G \times X \times X \to X$ 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.

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.

If I understand the question in the comment 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 $G \times X \times X \to X$ 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.

I tried to construct an example like the one Jason Starr was talking about (skimming Brian's webpage, I couldn't figure out which papers were supposed to be the scary ones). I think this works. Take $k$ to be a perfect field of characteristic $p$, with $p \neq 0$, $2$. Let $A = k[x,y]/(y^2-x^p)$. The normalization of $A$ is $\tilde{A} = k[t]$, with $y=t^p$ and $x=t^2$.

Let $G$ be the group scheme $\mathbb{G}_a$, so it has underlying space $k[\epsilon]/\epsilon^p$ and multiplication given by the map $\epsilon \to \epsilon \otimes 1 + 1 \otimes \epsilon$. If, like me, you prefer to think in terms of functors of points, $G(R) = \{ \epsilon \in R : \epsilon^p =0 \}$ and the multiplication map $G(R) \times G(R) \to G(R)$ is $(\epsilon_1, \epsilon_2) \mapsto \epsilon_1 + \epsilon_2$.

Let $G(R)$ act on $A(R)$ by $(\epsilon, (x,y)) \mapsto (x+2 \epsilon, y)$. If you prefer maps of algebras, $x \mapsto 1 \otimes x + 2 \epsilon \otimes 1$, $y \mapsto 1 \otimes y$. I claim that this action does not lift to $\tilde{A}$. Suppose, to the contrary, that $t \mapsto 1 \otimes t_0 + \epsilon \otimes t_1 + \cdots + \epsilon^{p-1} \otimes t_{p-1}$, with the $t_i \in k[t]$. Writing out that the action must preserve the relation $t x^{(p-1)/2} = y$ gives $$ \left( 1 \otimes t_0 + \epsilon \otimes t_1 + \cdots \right) (1 \otimes t^2 + 2 \epsilon \otimes 1)^{(p-1)/2} = 1 \otimes t^p $$

Equating the coefficients of $1$ and $\epsilon$ gives $t_0 t^{p-1} = t^p$ and $t_1 t^{p-1} + (p-1) t_0 t^{p-3} = 0$. So $t_0=t$ and $t_1 = 1/t$. But $1/t$ isn't in $k[t]$.

Morally, the action wants to be $(\epsilon, t) \mapsto t \sqrt{1+2\epsilon t^{-2}}=1+\epsilon t^{-1} - (1/2) \epsilon^2 t^{-3} + \cdots $. The trouble is that $\left( t \sqrt{1+2\epsilon t^{-2}} \right)^k$ is in $k[t, \epsilon]/\epsilon^p$ when $k$ is even or is $\geq p$, but not in general.

If I understand the question in the comment 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 $G \times X \times X \to X$ 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.