Unitary units in modular group algebras of 2-groups

We study the unitary subgroupV _?(F₂ G) in the group algebras F₂ Gof 2-groups of maximal class over the field F₂of two elements. We show that there does not exist a normal complement to Gin V _?(F₂ G) if and only if Gis the dihedral group of order 2ⁿ(n≥ 5) or the semidihedral group of order 2ⁿ(n≥5). We also describe V _?(F₂ G) when the subgroup Ghas order 8 and 16 and in this case there exist a normal complement of Gin V _?(F₂ G).