model categories - Equivariant map preserves stabilizer

Let $G$ be a group and $X$ a set equipped with a transitive right $G$-action. Further, let $c: Xto X$ be a $G$-equivariant map. Is it true that $text{Stab}(x) = text{Stab}(c(x))$ for all $xin X$?

This doesn't seem to be an interesting mathoverflow question on its own, but the reason I ask is the following: In Hovey's book on Model Categories Hovey proves some sort of five lemma for pointed model categories (Thm.6.5.3). In the end of the proof, he seems to conclude from $alphatext{Stab}(x)alpha^{-1}subsettext{Stab}(x)$ that equality holds; I don't understand how this can be done. The above statement comes from a try to fix this gap.

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