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

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 $\alpha\text{Stab}(x)\alpha^{-1}\subset\text{Stab}(x)$ that equality holds; I don't understand how this can be done. The above statement comes from a try to fix this gap.