On generalized quaternion Sylow $2$-subgroups and $2$-nilpotence of finite groups

In 2020, Mousavi proved that a finite group $G$ with a generalized quaternion Sylow $2$-subgroup $S$ is $2$-nilpotent if $3\nmid |G|$ or if $G$ is solvable and $|S|>16$. In this note, we generalized the result of Mousavi and provided a simpler proof. In detail, we showed that a finite group with a generalized quaternion Sylow $2$-subgroup is $2$-nilpotent if, and only if, it is $\mathrm{S}{\mathrm{L}}_2(3)$-free and that, for a finite group with a generalized quaterion Sylow $2$-subgroup of order strictly greater than $16$, the properties of being $2$-nilpotent, solvable, and $2$-constrained are equivalent.