I make a remark in the opposite direction. Let $p \geq 17$ be a Fermat or Mersenne prime, so that $X = {\rm PSL}(2,p)$ has a dihedral Sylow $2$-subgroup $D$ which is maximal. Let $d >1$ be a power of $2$, and let $Q$ be a transitive $2$-subgroup of $S_{d}$. Let $A= {\rm Aut}(X)$ and let $T$ be a Sylow $2$-subgroup of $A$. Then 
$T$ is a maximal subgroup of $XT$.

Using this action of $Q$ on $d$ points, form the wreath product $XT \wr Q$. Then $T \wr Q$ is a Sylow $2$-subgroup of $(XT) \wr Q$, and is a maximal subgroup of $(XT) \wr Q$.