We present the electrical resistivity data under application of pressures up to $sim$ 26 GPa and down to 50 mK temperatures on YbFe$_2$Zn$_{20}$. We find a pressure induced magnetic phase transition with an onset at $p_c$=18.2$pm$0.8 GPa. At ambient pressure, YbFe$_2$Zn$_{20}$ manifests a heavy fermion, nonmagnetic ground state and the Fermi liquid behavior at low temperatures. As pressure is increased, the power law exponent in resistivity, $n$, deviates significantly from Fermi liquid behavior and tends to saturate with $n$ = 1 near $p_c$. A pronounced resistivity maximum, $T_text{max}$, which scales with Kondo temperature is observed. $T_text{max}$ decreases with increasing pressure and flattened out near $p_c$ indicating the suppression of Kondo exchange interaction. For $p>p_c$, $T_text{max}$ shows a sudden upward shift, most likely becoming associated with crystal electric field scattering. Application of magnetic field for $p>p_c$ broadens the transition and shifts it toward the higher temperature, which is a typical behavior of the ferromagnetic transition. The magnetic transition appears to abruptly develop above $p_c$, suggesting probable first-order (with changing pressure) nature of the transition; once stabilized, the ordering temperature does not depend on pressure up to $sim$ 26 GPa. Taken as a whole, these data suggest that YbFe$_2$Zn$_{20}$ has a quantum phase transition at $p_c$ = 18.2 GPa associated with the avoided quantum criticality in metallic ferromagnets.
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