Invariants of classical braids valued in $G_{n}^{2}$

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 $G_{n}^{3}$.