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The sensitivity of the late stages of stellar core collapse to electron-capture rates on nuclei is investigated, with a focus on electron-capture rates on 74 nuclei with neutron number close to 50, just above doubly magic 78Ni. It is demonstrated that variations in key characteristics of the evolution, such as the lepton fraction, electron fraction, entropy, stellar density, and in-fall velocity are about 50% due to uncertainties in the electron-capture rates on nuclei in this region, although thousands of nuclei are included in the simulations. The present electron-capture rate estimates used for the nuclei in this high-sensitivity region of the chart of isotopes are primarily based on a simple approximation, and it is shown that the estimated rates are likely too high, by an order of magnitude or more. Electron-capture rates based on Gamow–Teller strength distributions calculated in microscopic theoretical models will be required to obtain better estimates. Gamow–Teller distributions extracted from charge-exchange experiments performed at intermediate energies serve to guide the development and benchmark the models. A previously compiled weak-rate library that is used in the astrophysical simulations was updated as part of the work presented here, by adding additional rate tables for nuclei near stability for mass numbers between 60 and 110. [ABSTRACT FROM AUTHOR]