The normalizer problem for finite groups with prescribed 2-subgroups

Suppose that $X$ is a finite group with prescribed $2$-subgroups. Under certain conditions, it is shown that the normalizer property holds for $X$. In particular, let $X$ be a semidirect product of a normal $2$-complement $O_{2^{\prime }}(X)$ by a Sylow $2$-subgroup $P$. If $m^3$ is conjugate to $m$ or $m^{-1}$, for all $m\in P$, then $X$ has the normalizer property. Our result generalizes a result due to Mazur, which states that the normalizer property holds for finite groups that have the Sylow $2$-subgroup of order $2$.