  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.
