In general are we just supposed to make an educated guess about what the $G$-set is for a group action is if it's not specified? Here are two examples of what I mean.
I am asked to find a fixed point for the dihedral group of the square. Typically when I think of group action on a square, I think that the symmetries are moving the vertices around. However, if I see it that way, then there is no fixed point for the square. Am I supposed to instead take the $G$-set to be the set of all points in the square, so that the origin is a fixed point?
Similarly, would the $G$-set for $S_{4}$ be the set $X = \{1,2,3,4\}$? So then the orbit of 1 is $\{1,2,3,4\}$ because, for instance, $i*1 = 1$ and $(12)*1 = 2$ and so on?
Given a group $G$ and $G$-set $X$, but where we are not told the $G$-action, in general you can't deduce how $G$ acts on $X$. Sometimes it should be clear from context: $S_4$ acts naturally on $\{1,2,3,4\}$, and $D_4$ acts naturally on the vertices of a square. However, we also about a $G$-action from $D_4$ on the vertices of a square by having each element act as the identity.
When we talk about fixed points on group actions, we normally talk about the fixed points of a single element of $G$. For example, the nontrivial rotations in $D_4$ have no fixed points.