  We provide some necessary and some sufficient conditions for the automorphism
group of a free product of (freely indecomposable, not infinite cyclic) groups
to have Property (FA). The additional sufficient conditions are all met by
finite groups, and so this case is fully characterised. Therefore this paper
generalises the work of Leder (arXiv:1810.06287) for finite cyclic groups, as
well as resolving the open case of that paper.
