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