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'}(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 $.