  We study the axial perturbations of spontaneously scalarized black holes in
Einstein-Gauss-Bonnet (EGB) theories. We consider the nodeless solutions of the
fundamental branch of the model studied in [1], which possesses a region of
radially stable configurations, as shown in [2]. Here we show that almost all
of the radially stable black holes are also stable under axial perturbations.
When the axial potential is no longer strictly positive, we make use of the
S-deformation method to show stability. As for the radial perturbations,
hyperbolicity is lost below a certain critical horizon size for a fixed
coupling constant. In the stable region, we determine the spectrum of the
quasinormal modes by time evolution and by solving the associated
time-independent eigenvalue problem.
