Forgiveness in Mourning for Distant Relatives

What is the law when the mourner forgives his children from the rule that all who mourn for him mourn with him, and does not want them to observe mourning at all?

The father can forgive, and they do not need to observe the laws that all who mourn for him mourn with him.

In the Gemara Moed Katan 20a, it is explained that there is no law of mourning for all who mourn for him except in the presence of his relative. The Rosh wrote: "And nowadays it is customary to be lenient in mourning for those who mourn with them, relying on the fact that it is only for the honor of the mourner," as Ramban wrote in "Torat HaAdam" and "Hagahot Maimoniot" chapter 2 (letter 2). And so ruled the Rema, that everyone is accustomed to forgive. However, he added that the custom, as written by the Terumat HaDeshen, is that all relatives of the mourner who are disqualified from testimony observe with him some of the laws of mourning. And the Pitchei Teshuva wrote there in S"K 4: "See in Shut Adnei Paz, siman 11, where it is written about one whose relative who is disqualified from testimony died elsewhere, and he has mourners here who do not yet know about it, that he does not need to change his clothes as is customary in the first Shabbat. Since the main custom is for the honor of the mourners. And so it implies in the Rema, who wrote for the honor of the mourners, if there are no mourners here, then certainly exempt, and even there is a prohibition due to the disrespect of the honor of Shabbat. And it seems to me that what the Rema wrote, and nevertheless it is customary, refers even if there are no relatives mourning here, and so it is explained in the Darkei Moshe, see there, if he needs to change his clothes." And therefore, according to the Rema, it is even not in the presence of the mourner, it is for the honor of the deceased.

And today it is customary to mourn only together with his father and mother, and only when mourning for their father and mother, therefore, certainly it is for the honor of his father and mother, and not for the honor of the deceased, and therefore only in that city, certainly the forgiveness of their father and mother is effective.