  It is not known whether or not the stable rational cohomology groups $\tilde
H^*(Aut(F_\infty);\Q)$ always vanish. We show that either the rational
cohomology does not vanish in certain dimensions, or the integral cohomology of
a moduli space of pointed graphs does not stabilize in certain other
dimensions. Similar results are stated for groups of outer automorphisms. This
yields that $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and
$H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli
spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and
$X_m$ of ``outer space'' and ``auter space'', respectively, introduced by
Culler and Vogtmann and by Hatcher and Vogtmann.
