  Asymptotically safe gravity is based on the idea of the dependence of the
gravitational coupling upon the distance from the origin, approaching its
classical value in the weak field regime. We consider three cases of
identifying the cut-off parameter in the asymptotically safe gravity, leading
to the three distinctive models for black holes. We find that the deviation of
the fundamental mode from the Schwarzschild limit is a few percent, in contrast
to the higher overtones, where the deviation reaches hundreds of percent, even
when the fundamental mode almost coincides with the Schwarzschild mode. This
behavior is connected with the fact that the quantum correction to the black
hole spacetime is strong near the event horizon, but quickly falls off with
distance and negligible near the peak of the effective potential surrounding
the black hole.
