  The present paper constructs unbounded quasimorphisms that are invariant
under all automorphisms on free products of more than two factors and on graph
products of finitely generated abelian groups. This includes many classes of
right angled Artin and right angled Coxeter groups. We discuss various
geometrically arising families of graphs as examples and deduce the
non-triviality of an invariant analogue of stable commutator length recently
introduced by Kawasaki and Kimura for these groups.
