CW structure induced by a group action

Let $X$ be a CW-complex on which a group $G$ acts. How does the CW-complex structure of $X/G$ relates to that of $X$?

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Wouldn't this depend a lot on the way the group acts on $X$? Like, is it effective, transitive, free, etc. etc.? –  Jon Beardsley Dec 7 '11 at 17:30
i don't know, for example $S^2$ has one 0-cell and one 2-cell and its quotient by $\mathbb Z_2$ has one $i$-cell for $i=0,1,2$. –  palio Dec 7 '11 at 17:39
The other thing is, there are lots of different CW structures you can put on a space, so this complicates it a little. –  Jon Beardsley Dec 7 '11 at 17:55