  The main result of this paper is that the outer automorphism group of a free
product of finite groups and cyclic groups is semistable at infinity (provided
it is one ended) or semistable at each end. In a previous paper, we showed that
the group of outer automorphisms of the free product of two nontrivial finite
groups with an infinite cyclic group has infinitely many ends, despite being of
virtual cohomological dimension two. We also prove that aside from this
exception, having virtual cohomological dimension at least two implies the
outer automorphism group of a free product of finite and cyclic groups is one
ended. As a corollary, the outer automorphism groups of the free product of
four finite groups or the free product of a single finite group with a free
group of rank two are virtual duality groups of dimension two, in contrast with
the above example. Our proof is inspired by methods of Vogtmann, applied to a
complex first studied in another guise by Krsti\'c and Vogtmann.
