  The aim of the present note is to enhance groups $G_{n}^{3}$ and to construct
new invariants of classical braids. In particular, we construct invariants
valued in $G_{N}^{2}$ groups. In groups $G_{n}^{2}$, the identity problem is
solved, besides, their structure is much simpler than that of $G_{n}^{3}$.
  I am grateful to Huyue Yan for pointing out a small mistake in the previous
version.
