The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ have the interesting property that $A$ and $B$ are non-isomorphic, but the point stabilizers $A_x$ and $B_y$ are isomorphic permutation groups (namely a Frobenius group of order 21 acting on 7 points).

Are there other examples of non-isomorphic two-transitive permutation groups with isomorphic point stabilizers?

Note that I really require the point stabilizers to be isomorphic as permutation groups, not just as abstract groups.

(My original motivation for this question is the fact that from every such pair of groups, one can construct an example of two non-isomorphic totally disconnected locally compact groups with isomorphic compact open subgroups. In the meantime, it is already known that there are many examples of this phenomenon, but at that time, this was unknown. I nevertheless believe that the question might still be interesting in its own right.)