Can One Who Publicly Desecrates Shabbat Complete a Minyan?

Hello, In cases of pressing need when it is impossible to assemble a minyan of Shabbat observers, it is permitted to include even a person who desecrates Shabbat, provided he has basic faith in the Creator of the Universe, shows respect for the religion, and has a fundamental understanding of the meaning of prayer and Kaddish. Nevertheless, they should not recite the repetition of the Amidah (chazarat hashatz); rather, the prayer leader should recite one aloud with the kedushah up to "HaKel HaKadosh," after which the prayers should continue quietly. 

Sources: See the Kitzur Shulchan Aruch, part 4 (Siman 55, paragraph 34), which states that in our times, public desecrators of Shabbat do so out of ignorance of Torah and the severity of the prohibition, and they may be included in a minyan in times of need. Nevertheless, they should not recite the repetition of the Amidah but only one aloud with the kedushah up to "HaKel HaKadosh," then continue quietly. Also see Piskei Teshuvot, part 1 (Siman 55, note 21), which adds that leniency in pressing times applies only if the person has basic faith in Hashem, respects the religion, and understands the significance of prayer and Kaddish. Blessings, Hillel Meirs