  Let $G=A \ast B$ be a free product of freely indecomposable groups. We
explicitly construct quasimorphisms on $G$ which are invariant with respect to
all automorphisms of $G$. We also prove that the space of such quasimorphisms
is infinite-dimensional whenever $G$ is not the infinite dihedral group. As an
application we prove that an invariant analogue of stable commutator length
recently introduced by Kawasaki and Kimura is non-trivial for these groups.
