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A renormalization group approach based on the idea that the primary contribution to the Schwarzschild-like black hole spacetime arises from the value of the gravitational coupling is considered. The latter depends on the distance from the origin and approaches its classical value in the far zone. However, at some stage, this approach introduces an arbitrariness in choosing an identification parameter. There are three approaches to the identification: the modified proper length (the Bonanno-Reuter metric), the Kretschmann scalar (the Hayward metric), and an iterative, and, in a sense, coordinate-independent procedure (Dymnikova solution). Using the Wentzel-Kramers-Brillouin method, gray-body factors are calculated for the Standard Model massless test fields and their corresponding energy emission rates. For all of these solutions, it is found that the intensity of Hawking radiation of massless fields is significantly suppressed by several or more orders once the quantum correction is taken into consideration. This indicates that the effect of suppression of the Hawking radiation may be appropriate to the quantum corrected black holes in asymptotically safe gravity in general and is independent on the particular choice of the identification parameter.