Let me start by admitting that my question is going to be somewhat vague. But hopefully it is one of these vague questions that can be immediately answered by an expert in the appropriate area.
Recall that the braid group $\mathfrak{B}_n$ on $n$ strands can be identified with the fundamental group the configuration space $X_n = S_n \backslash (\mathbf{C}^n - \Delta)$ of $n$ distinct points in the complex plane. Any local system on $X_n$ hence gives rise to a representation of $\mathfrak{B}_n$ via parallel transport along paths. A prominent example of such a local system being the Knizhnik-Zamolodchikov system.
The braid group has a variant, the framed braid group $\widehat{\mathfrak{B}_n}$ which can be identified with the semi-direct product $\mathbf{Z}^n \rtimes \mathfrak{B}_n$ algebraically, and geometrically as the group given by $n$ embedded ribbons modulo ambient isotopy, where $(z_1,\ldots ,z_n) \in \mathbf{Z}^n$ corresponds to the number of twists of the ribbons.
Now my question is: Is there a canonical way to associate to local systems on $X_n$ representations of $\widehat{\mathfrak{B}_n}$, other than by pulling back the representation of $\mathfrak{B}_n$ along $\widehat{\mathfrak{B}_n} \twoheadrightarrow \mathfrak{B}_n$?
