A question about the involution in simple groups. Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question which the first one includes the second.


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*Is it possible that centralizer of an involution in $G$ be a $2$-group?

*Is it possible that centralizer of a central involution of $S$ in $G$ be a $2$-group?

 A: Yes, though it is rare.
Lemma: Suppose $q=2^n \pm 1 > 3$ is a prime power. Then the simple group $\operatorname{PSL}(2,q)$ has a single class of involutions, and its centralizer is the Sylow 2-subgroup, a dihedral group of order $2^n$.
Example: PSL(2,7) = GL(3,2), PSL(2,9) = A6, PSL(2,17), PSL(2,31), PSL(2,127), PSL(2,257).
Lemma: The Suzuki groups have a single class of involutions, and its centralizer is the Sylow 2-subgroup (a specific type of Suzuki 2-group, necessarily non-abelian). Suzuki (1961)
Example: Sz(8), Sz(32)
Also there are a few weirdos:
Example: PSL(3,4) has a single class of involutions, and its centralizer is the Sylow 2-subgroup. PSp(4,4) has three classes of involutions (all central), and ONE of the classes has centralizer a Sylow 2-subgroup (the others contain the Sylow as index 15).
There are no other examples of order less than 300,000,000 of simple groups with non-abelian Sylow 2-subgroups and an involution whose centralizer is a 2-group. PSp(4,8) is similar to PSp(4,4). I haven't checked if it fits in an infinite family.
There are no examples in odd characteristic and high-rank: checking the table in  this answer, the last column on page 172 and 174 includes a subgroup of the centralizer, and except for A1, it is not even a solvable group, much less a 2-group.
Suzuki (1961) classified the groups in which every involution's centralizer is a 2-group. The classification is particularly clear for non-abelian simple groups: $\operatorname{PSL}(2,p)$ for a $p$ a Fermat or Mersenne prime, $\operatorname{PSL}(2,9)$, $\operatorname{PSL}(3,4)$, $\operatorname{PSL}(2,2^n)$, or $\operatorname{Sz}(2^{2n+1})$. Note the $\operatorname{PSp}(4,2^n)$ don't appear here, since they have some good involutions and some bad ones.


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*Suzuki, Michio.
“Finite groups with nilpotent centralizers.”
Trans. Amer. Math. Soc. 99 (1961) 425–470.
MR131459
DOI:10.2307/1993556
