I am looking for a finite, transitive and nonregular permutation group $G$ acting on $\Omega$, such that every nontrivial element fixes at most two points and such that
i) the point stabilizers $G_{\alpha}$ have even order,
ii) $G$ has even degree, i.e. $|\Omega|$ is even,
iii) the Sylow $2$-subgroups are dihedral or semidihedral
iv) $|S_{\alpha}| > 2$ for some Sylow $2$-subgroup $S$ and $\alpha \in \Omega$.
The condition i) and ii) [and also iv)] imply that four divides $|G|$. For example if $G = \mathcal S_4$ in its natural action, then i), ii) and iii) are fulfilled (see for example here), but as for example for $\alpha = 1$ we have $G_{\alpha} = \langle (234), (23), (34), (24) \rangle$ and $S = \langle (1234), (24) \rangle$, then $S \cap G_{\alpha} = \{ (), (24) \}$, and by symmetry considerations we see that every Sylow $2$-subgroup intersects with the stabilizers in a subgroup of order two, so this is not an example.
So okay do you know any examples? (Remark: I added the GAP-tag, maybe some clever GAP-user knows how to use a computer program to find an example).
${\rm PSL}(2,q)$ with $q \equiv 1 \bmod 8$ or ${\rm PGL}(2,q)$ with $ q \equiv 1 \bmod 4$ are examples with dihedral Sylow $2$-subgroups. So ${\rm PGL}(2,5)$ is the smallest example.
For some $q$, such as $q=9$, there is a related example ${\rm PSL}(2,q^2).\langle \tau \rangle$, where $\tau$ induces a product of a field and a diagonal isomorphism, with semidihedral Sylow $2$-subgroups.