Twin groups representations

We construct two representations of the twin group $T_n, n\geq 2$, namely $\eta_1: T_n \rightarrow \text{Aut}(\mathbb{F}_n)$ and $\eta_2: T_n \rightarrow \text{GL}_n(\mathbb{Z}[t^{\pm 1}])$, where $\mathbb{F}_n$ is a free group with $n$ generators and $t$ is indeterminate. We then analyze some characteristics of these two representations, such as irreducibility and faithfulness. Moreover, we prove that both representations can be extended to the virtual twin group $VT_n$ in the $2$-local extension way, for $n\geq 2$, and we find their $2$-local extensions. On the other hand, we obtain a different result for the welded twin group $WT_n$. More deeply, we show that $\eta_1$ cannot be extended to $WT_n$ in the $2$-local extension way, for $n\geq 3$, while $\eta_2$ can be extended to $WT_n$ in the $2$-local extension way, for $n\geq 2$, and we find its $2$-local extensions.