I need help with this question.
Suppose that G is a group acting on a set of objects S, and that C is the set of colorings of elements of S using the colors in a set R. Let $\overline c$ denote the orbit of c in C with respect to the action of G. Let $\rho$ be a permutation acting on R. Prove that $\{ c \in C: \rho^ * (c) \in \overline c \} = \bigcup_{\rho^ * ( \overline c )= \overline c} \overline c$.
The textbook, while mentioning the proof of Generalization of Pólya Enumeration Theorem, says this. Since G is a group, it is straightforward to show that the set of all colorings that are invariant under ρ is the union of all the orbits $\overline c$ where ρ($\overline c$) = $\overline c$.
Here is what I have tried so far. My approach is as follows. It is sufficient to show that if ρ(c) $\in \overline c$ then for all c' $\in \overline c$ ρ(c') will be in $\overline c$. This is where I am having a hard time reasoning it out. The equivalence classes $\overline c$ are in relation to the group G and its permutations $\pi$ and NOT group R (or any such ρ). How exactly do I use the hint that G is a group to solve this problem? Thanks in advance!