Coxeter quotients of the automorphism group of a Coxeter group

We show that for a large class $\mathcal{W}$ of Coxeter groups the following holds: Given a group $W_{\mathrm{\Gamma }}$ in $\mathcal{W}$, the automorphism group ${\rm Aut}(W_\Gamma)$ virtually surjects onto some infinite Coxeter group. In particular, the group $\mathrm{A}\mathrm{u}\mathrm{t}(W_{\mathrm{\Gamma }})$ is virtually indicable and therefore does not have Kazhdan’s property (T).