Let $G$ be a group of symmetries of a Coxeter system $(W,S)$ and let $f:W \to GL (V)$ be the linear representation associated with a root basis $(V, \langle .,. \rangle, \Pi)$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde \Pi \subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde \Pi)$ is a root basis of $W^G$, and we show that the induced representation $f^G : W^G \to GL( V^G)$ is the linear representation associated with $(V^G, \langle .,. \rangle, \tilde \Pi)$. In particular, the latter is faithful.