The results of a study of the beta decays of three proton-rich nuclei with $T_z=text{-}2$, namely $^{48}$Fe, $^{52}$Ni and $^{56}$Zn, produced in an experiment carried out at GANIL, are reported. In all three cases we have extracted the half-lives and the total $beta$-delayed proton emission branching ratios. We have measured the individual $beta$-delayed protons and $beta$-delayed $gamma$ rays and the branching ratios of the corresponding levels. Decay schemes have been determined for the three nuclei, and new energy levels are identified in the daughter nuclei. Competition between $beta$-delayed protons and $gamma$ rays is observed in the de-excitation of the $T=2$ Isobaric Analogue States in all three cases. Absolute Fermi and Gamow-Teller transition strengths have been determined. The mass excesses of the nuclei under study have been deduced. In addition, we discuss in detail the data analysis taking as a test case $^{56}$Zn, where the exotic $beta$-delayed $gamma$-proton decay has been observed.
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