This is exercise 8B.7 in Isaacs's Finite Group Theory. Let $G$ be a primitive permutation group on $\Omega$, and let $H$ be the stabilizer of $\alpha\in\Omega$. Suppose $H$ has an orbit of size $3$ on $\Omega-\alpha$ (say on the 3 points $\beta$,$\gamma$,$\delta$), and define $D$ to be the stabilizer of $\beta$ in $H$ (so that $D$ is the pointwise stabilizer of $\lbrace\alpha,\beta\rbrace$).
I 'm trying to show $D$ is a 2-group; the hint in FGT is to let $K=core_H(D)$ and show $O^2(D)=O^2(K)$. I have done this. Now I need to show $O^2(K)=1$; most likely I need to show $O^2(K)$ is normal in $G$. So I need some $g\in G-H$ normalizing $O^2(K)$. Probably I can find some $g\in G$ permuting $\lbrace\alpha,\beta,\gamma,\delta\rbrace$, so that $K^g=K$.
So does anyone know how to finish the exercise?