Although many random-phase approximation (RPA) calculations of the Gamow-Teller (GT) response exist, this is not the case for calculations going beyond the mean-field approximation. We apply a consistent model that includes the coupling of the GT resonance to low-lying vibrations, to nuclei of the $fp$ shell. Among other motivations, our goal is to see if the particle-vibration coupling can redistribute the low-lying GT$^+$ strength that is relevant for electron-capture processes in core-collapse supernova. We conclude that the lowering and fragmentation of that strength are consistent with the experimental findings and validate our model. However, the particle-vibration coupling cannot account for the quenching of the total value of the low-lying strength.
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