Let $A$ be ring endowed with $G$-action by a group. Take any arbitrary $A$-module $M$.
Is there a canonical way to extend the $G$-action to $M$? I'm not sure how to avoid the problem with well definess if I try to settle the action in naive way via $am \mapsto a^g m$.
The background of my question is Automorphism Group of a Variety acts on Local Sections
No, there is not. For example, in the case where $M$ is $A$ itself, in order for the action of $G$ on $M$ to deserve the name "canonical", it would need to commute with any automorphism of $M$ as a module. Concretely, that means that every unit of $A$ would have to be fixed by the action of $G$, which certainly is not true in general.