It is a fact that the symmetric groups have as many 2-Sylow subgroups as possible. More precisely, for all $n \geq 1$, the number of 2-Sylow subgroups in $S_n$ is exactly $n!/2^{\nu_2(n!)}$, which is the index of a 2-Sylow subgroup of $S_n$. This follows from (or, depending on which direction you're coming from, proves) the fact that one (equivalently, all) 2-Sylow subgroup is self normalizing.
It isn't too hard to show that given a prime $p$, there is a family of finite groups $(G_n)$ such that $\nu_p(|G_n|) \rightarrow \infty$ and all the $p$-Sylow subgroups of $G_n$ are self-normalizing.
I want to generalize this to two primes in the obvious way, but I am encountering difficulty. The following would be a good start.
Given distinct primes $p$, $q$, does there exist a finite group $G$ such that $pq \ \Big| \ |G|$ and all $p$-Sylow, all $q$-Sylow subgroups are self-normalizing?
