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Using the general parametrization of spherically symmetric and asymptotically flat black holes in arbitrary metric theories of gravity and implying that: a) the post-Newtonian constraints are taken into account and b) basic astrophysically relevant characteristics (such as, dominant quasinormal modes, frequency at the innermost stable circular orbit, binding energy, radius of the shadow etc.) are indistinguishable from their Schwarzschild values, we propose a simple metric which depends on three independent parameters (coefficients of the parametrization). Variation of these three parameters can, nevertheless, lead to the two distinctive features. The first is the black-hole temperature, and consequently the Hawking radiation, which can differ a lot from its Schwarzschild limit. The second is the outburst of overtones which become extremely sensitive to small changes of the parameters.