Is the divorced father obliged to pay Zakat al-Fitr for his children who are in their mother's custody, or is it incumbent upon the mother?

A father must spend on his children and pay zakat (definition) al-fitr on their behalf. He may authorize the mother to pay it, either from her own money if she donates it, or by giving her the zakat money in addition to the maintenance expenses, so she can deliver it to those who are eligible. The father is obligated to pay the fitra for his well-off child. However, if the child himself is well-off, then his maintenance and fitra are from his own wealth, not from his father.