Let $G$ be a transitive, nonregular permutation group acting on $\Omega$. Suppose that $|\Omega|$ is odd, then $G_{\alpha}$ contains a full Sylow $2$-subgroup $S$ of $G$. Suppose that $G = G_{\alpha}\cdot O(G)$, where $O(G)$ denotes the largest normal subgroup of odd order (the so called $2'$-core of $G$). Now suppose $1 \ne X < G_{\alpha}$ is a proper nontrivial subgroup of $G_{\alpha}$ and assume $X \le G_{\alpha}\cap G_{\beta}$ where $1 \ne G_{\alpha}\cap G_{\beta} \ne G_{\alpha}$.
Then is it true that $X$ normalizes $S$?
This is used in a proof, where it is said it could be assumed without loss that $X$ normalizes $S$ by coprime action. But I do not see what should act coprimely on another set, the only ones I see is that $S$ acts coprime on $O(G)$, but this does not help me in seeing the relation between $X$ and $S$?