2 added stuff on other Sylow subgroups; added 2 characters in body

For $q$ odd, it is not difficult to describe the structure of a Sylow 2-subgroup of $G={\rm GL}(2,q)$GL}(n,q)$, so let me do that. If$q \equiv 1 \bmod 4$, then the subgroup of monomial matrices of$G$contains a Sylow$2$-subgroup of$G$, which is a wreath product of a cyclic group of order$t$(where$t$is the 2-part of$q-1$), and a Sylow 2-subgroup of the symmetric group$S_n$. The Sylow subgroups of$S_n$are of course themselves built up as wreath products of cyclic groups. It is a little more complicated when$q \equiv 3 \bmod 4$. In that case, let$t$be the 2-part of$q^2-1$. Then, if$n$is even, a Sylow 2-subgroup of$G$is a wreath product of a Sylow 2-subgroup of${\rm GL(2,q)}$, which is semidihedral of order$2t$, with a Sylow 2-subgroup of$S_{n/2}$. If$n$is odd, then it is a direct product of a Sylow 2-subgroup of${\rm GL}(n-1,q)$with a cyclic group of order 2. To get a Sylow 2-subgroup of${\rm PGL}(n,q)$, you have to factor our the cyclic scalar subgroup, which is a diagonal of the base group of the wreath product. For$q$even, the upper unitriangular matrices form a Sylow 2-subgroup of${\rm PGL}(2,q)$PGL}(n,q)$.

I don't know much about Schur indices but I did some quick computations in Magma, and I found that for Sylow 2-subgroups of ${\rm GL}(n,q)$ for small $n,q$, going up $(n,q)=(6,5)$ and $(8,3)$, the Schur indices of all irreducible representations are indeed 1, which seems to provide good experimental evidence for your conjecture. You might find it easier to try and prove it for ${\rm GL}(n,q)$, since the structure of the Sylow 2-subgroups can be described so explicitly, at least for odd $q$.

Added later. For odd prime $r$ not dividing $q$, let $d$ be minimal such that $r$ divides $q^d-1$, let $t$ be the $r$-part of $q^d-1$, and let $m = \lfloor n/d \rfloor$. Then a Sylow $r$-subgroup of ${\rm GL}(n,q)$ is a wreath product of a cyclic group of order $t$ with a Sylow $r$-subgroup of $S_m$.

For all of the classical groups, Sylow subgroups in coprime characteristic arise in a similar way, as wreath products of a group coming from a base case with a Sylow subgroup of a symmetric group. But of course there are lots of little complications for the individual types. I had a student a few years ago write Magma code to construct all of these, so I can produce very explicit descriptions!

1

For $q$ odd, it is not difficult to describe the structure of a Sylow 2-subgroup of $G={\rm GL}(2,q)$, so let me do that.

If $q \equiv 1 \bmod 4$, then the subgroup of monomial matrices of $G$ contains a Sylow $2$-subgroup of $G$, which is a wreath product of a cyclic group of order $t$ (where $t$ is the 2-part of $q-1$), and a Sylow 2-subgroup of the symmetric group $S_n$. The Sylow subgroups of $S_n$ are of course themselves built up as wreath products of cyclic groups.

It is a little more complicated when $q \equiv 3 \bmod 4$. In that case, let $t$ be the 2-part of $q^2-1$. Then, if $n$ is even, a Sylow 2-subgroup of $G$ is a wreath product of a Sylow 2-subgroup of ${\rm GL(2,q)}$, which is semidihedral of order $2t$, with a Sylow 2-subgroup of $S_{n/2}$. If $n$ is odd, then it is a direct product of a Sylow 2-subgroup of ${\rm GL}(n-1,q)$ with a cyclic group of order 2.

To get a Sylow 2-subgroup of ${\rm PGL}(n,q)$, you have to factor our the cyclic scalar subgroup, which is a diagonal of the base group of the wreath product.

For $q$ even, the upper unitriangular matrices form a Sylow 2-subgroup of ${\rm PGL}(2,q)$.

I don't know much about Schur indices but I did some quick computations in Magma, and I found that for Sylow 2-subgroups of ${\rm GL}(n,q)$ for small $n,q$, going up $(n,q)=(6,5)$ and $(8,3)$, the Schur indices of all irreducible representations are indeed 1, which seems to provide good experimental evidence for your conjecture. You might find it easier to try and prove it for ${\rm GL}(n,q)$, since the structure of the Sylow 2-subgroups can be described so explicitly, at least for odd $q$.