Kaddish for a Father Who Left the Faith

If a father was Haredi and, unfortunately, left the faith, should his son say Kaddish for him after his death?

One who has strayed and left has the status of an apostate. The Rema in Siman 376, section 4, writes: "There is an opinion that if an apostate was killed by idolaters, his sons say Kaddish for him (even though the Rema in Siman 340, section 5, writes that one does not mourn for him, Kaddish is different, as Beit Hillel writes there. It seems that the reason is that one does not mourn for him because there is no sorrow over his death since he caused a great desecration of the Name in this world, but one says Kaddish for him because he has atonement since he was killed by idolaters)." The Shach in S"K 15 writes: "Only if killed, but if he died naturally, then not," and I wrote above in Siman 340 that if killed, he has atonement. However, the Gilyon Maharsha points out to the Rema from the response of the Radach (which is the response to the words of the Rema) that Kaddish is said for an apostate, and only if there is another mourner, the mourner for the apostate does not say, because saying Kaddish is an act of honoring the father, and the son of the apostate is not so obligated to honor him because he is wicked, but if there is no other mourner, the son of the apostate must say Kaddish. So ruled the Chatam Sofer in Shu"t Siman 326. And so ruled the Ktav Sofer in Siman 109. Therefore, he has the right of precedence. And saying Kaddish for twelve consecutive months is because the Rema in Siman 376, section 4, writes: "It is customary that Kaddish and prayer are said only for eleven months, so as not to make their father and mother wicked, for the judgment of the wicked is twelve months." And the Pitchei Teshuva in S"K 10 writes: "See in the book Chomot Yerushalayim Siman 257, where it is written that one who knows that his father and mother were wicked from those who are judged for twelve months should and is obligated to say Kaddish for twelve months." And so it is written in Shu"t Chatam Sofer YD 326. And see Shu"t Beit Yotzer OH Siman 45.