  For a large class of groups, we exhibit an infinite-dimensional space of
homogeneous quasimorphisms that are invariant under the action of the
automorphism group. This class includes non-elementary hyperbolic groups,
infinitely-ended finitely generated groups, some relatively hyperbolic groups,
and a class of graph products of groups that includes all right-angled Artin
and Coxeter groups that are not virtually abelian. This was known for $F_2$ by
a result of Brandenbursky and Marcinkowski, but is new even for free groups of
higher rank, settling a question of Mikl\'os Ab\'ert. The case of graph
products of finitely generated abelian groups settles a question of Michal
Marcinkowski. As a consequence, we deduce that a variety of Aut-invariant norms
on such groups are unbounded.
