I'm coming back to Group Theory after 20+ years out of it, and thus my vocabulary is a bit rusty. I'll try to express my problem the best I can.
Consider a group G acting on a set X. What I want is to let some elements of X to be indistinguishable. My guess is that it should be some quotient, but I don't know how to exactly define it.
For example, I have the group $G$ generated by $a=(1,2,3,4)$ and $b=(2,7,6,5)$. That's two circles intersecting at point 2 that can only rotate. I can express any element of $G$ as a word of $a$ and $b$. My problem is:
Now imagine points 1, 3 and 4 being of color blue. I want to express an element of $G$ as a word of $a$ and $b$ but I don't care where 1, 3 and 4 move as long as they fall on a blue position again. That is: elements 1, 3, and 4 are indistinguishable as far as color matters.
How can I define that? Am I talking about stabilizers? Isotropy? Some kind of quotient? Where can I read further to find what I am looking for?