Hall $\pi$ subgroup of subgroup of $G$ exist? Suppose that $G$ has a Hall-$\pi$ subgroup for a fixed prime set $\pi$. If $K$ is a subgroup of $G$ then can we say that $K$ has a Hall-$\pi$ subgroup ?

$G = {\rm PSL}(2,16)$ has a Hall $\\{3,5\\}$-subgroup (of order $15$), but $A_5 \cong {\rm PSL}(2,4) < G$ and $A_5$ has no such subgroup.

Another example: $K = {\rm PSL}(2,7) \cong {\rm PSL}(3,2)$ has no Hall $\\{2,7\\}$-subgroup, but $K < G={\rm PSL}(3,8)$, and $G$ does have one.

On the basis of two examples, I will make the wild conjecture that for any finite group $K$ and any set of primes $\pi$, $K$ embeds in a finite group that has a Hall $\pi$-subgroup.