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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 per cent, in contrast to the higher overtones, where the deviation reaches hundreds of per cent, 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 is negligible near the peak of the effective potential surrounding the black hole.