  In the paper, groups $\Gamma_n^4$ closely connected with braid groups are
researched from algebraic point of view. More exactly, for $n\geqslant7$, it is
proved that $\Gamma_n^4$ is a nilpotent finite $2$-group with $4$-torsion and
that its subgroup $(\Gamma_n^4)'$ is central.
