It is a theorem of Alperin, Feit and Thompson [Isaacs Character Theory book (4.9)] that if $G$ is a $2$-group in which the number of involutions is congruent to $1$ modulo $4$, then $G$ is cyclic or $|G:G'|=4$. In the latter case $G$ is dihedral, semidihedral or generalized quaternion.


Are there any results of a similar nature which hold for all finite groups? I want a result that says that in a finite group the number of involutions, modulo $4$, influences the structure of a Sylow $2$-subgroup of the group.



Number of involutions in a finite group modulo $4$
EDIT: Actually, there is an older reference which will give the result below. See



> 
> Herzog, Marcel
>  Counting group elements of order $p$ modulo $p^2$.
>  Proc. Amer. Math. Soc. 66 (1977), no. 2, 247–250.
> 
> 
> 


where you will also find references to other related papers.


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The following paper seems to give what you are looking for.



> 
> Murai, Masafumi
>  On the number of p-subgroups of a finite group.
>  J. Math. Kyoto Univ. 42 (2002), no. 1, 161–174. [link](https://projecteuclid.org/euclid.kjm/1250284716)
> 
> 
> 


Let $G$ be a finite group and fix a prime $p$. Denote by $s\_e(G)$ the number of subgroups of order $p^e$ in $G$. If $p^e$ divides the order of $G$, a classical result of Frobenius states that $s\_e(G) \equiv 1 \mod{p}$. The paper of Murai above studies how $s\_e(G) \mod{p^2}$ depends on the structure of a Sylow $p$-subgroup of $G$.


For $p = 2$, Theorem D in the paper implies the following:



> 
> Let $G$ be a finite group with a Sylow $2$-subgroup $P$ of order $2^n$ ($n \geq 2$). If $P$ is cyclic, dihedral, generalized quaternion, or semidihedral, then $s\_1(G) \equiv 1 \mod{4}$. Otherwise $s\_1(G) \equiv 3 \mod{4}$.
> 
> 
> 


Note that in this case $s\_1(G)$ is the number of involutions in $G$.


