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Nov 5, 2014 at 19:39 comment added Scott Andrews That certainly settles things, thanks for the link.
Nov 5, 2014 at 19:18 history edited Geoff Robinson CC BY-SA 3.0
added comment on case $q = p$.
Nov 5, 2014 at 18:38 comment added ahulpke for $n=2$ the paper `Computing Hall subgroups of finite groups', dx.doi.org/10.1112/S1461157012001039 by B.Eick and myself has a classification (Theorem 13)
Nov 5, 2014 at 17:35 history edited Geoff Robinson CC BY-SA 3.0
added another exception.
Nov 5, 2014 at 15:43 comment added Scott Andrews Thanks for the response! I still have to work out why $GL(2,q)$ does not have such a Hall subgroup for most $q$, but this reduction of the problem is very helpful.
Nov 5, 2014 at 15:42 vote accept Scott Andrews
Nov 5, 2014 at 14:59 history answered Geoff Robinson CC BY-SA 3.0