  Each pointed topological space has an associated $\pi$-module, obtained from
action of its first homotopy group on its second homotopy group. For the
$3$-ball with a trivial link with $n$-components removed from its interior, its
$\pi$-module $\mathcal{M}_n$ is of free type. In this paper we give an
injection of the (extended) loop braid group into the group of automorphisms of
$\mathcal{M}_n$. We give a topological interpretation of this injection,
showing that it is both an extension of Artin's representation for braid groups
and of Dahm's homomorphism for (extended) loop braid groups.
