What is the religious ruling on specifying a father's visitation rights for his حضانة (custody) child, and is the mother's specification of visitation days per week or during school holidays considered an injustice to the father?

If an agreement regarding the child's custody can be reached, it is preferable. If not, the competent authority will settle the disputes, because a judge's ruling removes disagreement, and the best interest of the child may be considered in custody matters.

The basic principle is that custody is a right of the mother unless she remarries or there is an impediment that nullifies her right, in which case custody transfers to the one who is most deserving of the child. If the spouses separate and one of them moves to another country, the father has a greater right to custody, provided there is a suitable female available to provide care in his household.

The father has the right to see his child, and if the parents dispute, the Islamic judge is the one who will decide. If the child is young and cannot do without his mother, then according to some scholars, custody belongs to the mother, in consideration of the child's best interest.