Is there any finite nonabelian simple group having nilpotent Hall $\pi$-groups with $|\pi|\geq 2$?

Question

Is there any finite nonabelian simple group having nilpotent Hall $\pi$-groups with $|\pi|\geq 2$ ?

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I wonder this because if a maximal subgroup $M$ of nonabelian simple group $G$ is nilpotent then It is known that $M$ is a Sylow subgroup of $G$.

Answer 1

The groups ${\rm PSL}(2,q)$ have cyclic Hall subgroups with this property whenever $q-1$ or $q+1$ is divisible by more than one odd prime.

For example ${\rm PSL}(2,29)$ and ${\rm PSL}(2,31)$ both have cyclic Hall subgroups of order $15$.