  We generalize an earlier result of Segev, which shows that {\em some\/}
component in a minimal counterexample to Quillen's conjecture must admit an
outer automorphism. We show in fact that {\em every\/} component must admit an
outer automorphism. Thus we transform his restriction-result on components to
an elimination-result: namely one which excludes any component which does not
admit an outer automorphism. Indeed we show that the outer automorphisms
admitted must include $p$-outers: that is, outer automorphisms of order
divisible by $p$. This gives stronger, concrete eliminations: for example if
$p$ is odd, it eliminates sporadic and alternating components -- thus reducing
to Lie-type components (and typically forcing $p$-outers of field type). For $p
= 2$, we obtain similar but less restrictive results. We also provide some
tools to help eliminate suitable components that do admit $p$-outers in a
minimal counterexample.
