Let $G$ a group, $X$ a set. An action of $G$ on $X$ is given by a mapping $(g,x) \to g \cdot x$, which satisfies $g \cdot (h \cdot x) = (gh) \cdot x$ and $e_G \cdot x = x$. In other words, to determine an action one must know how every element of the group acts on every element of the set.
Can an action be determined by means other then giving explicitly the action of every element of $G$ on every element of $X$?
What if I have several equations (depending on elements of X and G, and the action of G) that give some information of the action - when are the equations strong enough so they induce an action on the entire set? Furthermore, are there cases where I can fully recover the action?
For example, suppose that $X$ is a (given) set of elements of a field extension $K/k$, and $G \subseteq Gal(K / k)$. We would like to extend that action of $G$ to $K(X)$, such that the extended action satisfies several equations.
Is there any theory in mathematics that asks and answers this sort of questions?