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The cross sections for the elementary processes involving a change of the charge states of both particles during the interaction of He2+ ions with fullerene molecules are for the first time measured over a broad energy range of electron-volt energies. It is found that processes involving the capture of one or two electrons by the He2+ ions are accompanied by additional ionization of the fullerene and that the collisional contribution of the transfer-ionization processes increases with increasing velocity. Single-electron capture is rarely accompanied by fragmentation of the fullerene. Double-electron capture leads, with a higher probability, to fragmentation with the formation of several light charged fragments and, with a smaller probability, to fragmentation with the formation of a heavy charged fragment containing an even number of carbon atoms and light fragments in an uncharged state.