The Gemara in Berakhot 58b establishes that a deceased person is forgotten from the heart only after twelve months, deriving this from Tehillim 31:13, and Moed Katan 21b correspondingly treats twelve months as the full period of bereavement for a parent—the framework within which the kaddish question is set.
The custom among Ashkenazim is to recite kaddish for only eleven months rather than the full twelve, so as not to imply that one's parent requires the judgment reserved for the wicked, since the punishment of the wicked in Gehinnom runs for twelve months; Kitzur Shulchan Arukh 26:17 specifies that if the death occurred on, for example, the tenth of Shevat, kaddish ceases on the ninth of Tevet, because the tenth of Tevet would already constitute one day of the twelfth month, and a single day of a month counts as a full month, making it as though twelve months were observed, and Peninei Halakhah, Prayer 4:6 attributes this Ashkenazic practice to the Rema in Yoreh Deah 376:4.
The Sephardic custom, as recorded by Peninei Halakhah, Prayer 4:6, is to pause kaddish for only one week at the start of the twelfth month and then resume until the end of the year, and Ben Ish Hai, Halachot 1st Year, Vayechi 14 cites the Chida's ruling—grounded in the teaching of the Ari z"l that kaddish benefits the soul by elevating it—that kaddish should properly be recited for the full twelve months, with only the final week omitted so as to satisfy the widespread concern about implying wickedness.
Maamar Mordechai Limot HaChol 12:33 records that according to the view of the Ari, kaddish may be said even during the last week of the twelfth month, but the prevailing custom is to subtract one week of that month, and Maamar Mordechai Limot HaChol 12:34 adds that the Sephardic practice is to pause for one week at the start of the twelfth month and then resume for three more weeks until the year's end, while kaddish recited after a Torah study session—which any person may say—may be said by mourners even during the week of interruption.
Peninei Halakhah, Prayer 4:6 further notes that for one known to have been wicked—such as one who took his own life or was an apostate—kaddish is recited for the full twelve months, and Mishnah Berurah 132 underscores the great benefit kaddish confers on the deceased, recalling the famous account involving Rabbi Akiva, and notes that the kaddish after Aleinu was specifically assigned to mourners for this reason.
B'Mareh HaBazak Volume VI 75 addresses a practical complication: the Ashkenazic custom ends kaddish after eleven months from the day of burial, and when burial was delayed by more than one day after death, there is a dispute among the poskim whether the eleven months are counted from the day of death or the day of burial, with the ruling that each community should follow its own practice; when burial took place the day after death, the twelve-month bereavement period is calculated from the day of death.
Nitei Gavriel, Aveilus vol 2, perek 40 is not available in text for Mekoros to read, but our research indicates that he addresses the Rema's ruling in Yoreh Deah 376:4 that kaddish is recited for a parent throughout the entire year, including Sabbaths and festivals, in this source.
Minchas Yitzchak 9:42 is not available in text for Mekoros to read, but our research indicates that he addresses the apparent tension within the Rema's own words—one passage indicating eleven months and another indicating twelve months—and explains how both customs are reconciled in this source.
Nitei Gavriel, Aveilus vol 2, perek 41 is not available in text for Mekoros to read, but our research indicates that he addresses detailed halakhic rulings and customs regarding the recitation of kaddish in this source.
Modern poskim have addressed this question directly. Our research indicates that Shevet HaLevi (ח"ח סי' קס"ג שאלה ג') is relevant.
