The group of automorphisms of a group acts naturally on the space of quasimorphisms by precomposition. In 2023, Fournier-Facio and Wade proved that for a large class of groups there exists an infinite- dimensional space of quasimorphisms invariant for this action. Since their construction is non-explicit, it makes sense to ask whether some interesting subspaces of quasimorphisms admit or not fixed points for the action above. We will focus our attention on de Rham quasimorphisms, which were introduced by Barge and Ghys in the 80s. In this case, the (outer) automorphisms coincide with the (extended) mapping class group. We will prove that there are no non-trivial subspaces of de Rham quasimorphisms which are invariant for this action.