When does a divorced woman have custody of her children?
Question

                I know a sister who has divorced. She and her ex have two children, who haven't reached the age of 7 years old. We have heard, that in case she remarries, the father automatically will have the custody of the children. I would like to know if there are cases in which the mother does have the right to keep the children anyway.
Praise be to Allah.
      The mother has more right to her children – if they are 
      below the age of discernment – that her husband, so long as she does not 
      get married. If she gets married then the children go back to their 
      father. 

    Once the children reach the age of discernment, they should 
    be given the choice between their parents. 

    This applies if the father and mother are equal in terms of 
    religious commitment and good character. If one of them is of bad character 
    and not very religiously committed, he or she should not be given the right 
    of custody, because what counts is what is in the best interests of the 
    child, and his interests cannot be served with one who will neglect the 
    child’s rights. This is what is indicated by the hadeeths and this is what 
    was mentioned in fatwas by the most prominent scholars. 

    It was narrated from ‘Abd-Allah ibn ‘Amr (may Allah be 
    pleased with him) that the Prophet (blessings and peace of Allah be upon 
    him) said to a woman who sought custody of her child from her husband: “You 
    are more entitled to him so long as you do not get married.” Narrated by Abu 
    Dawood in his Sunan (2276); classed as hasan by Shaykh al-Albaani in
    al-Silsilah al-Saheehah (368). 

    Ibn al-Qayyim (may Allah have mercy on him) said:

    I heard our shaykh, may Allah have mercy on him – meaning, 
    Ibn Taymiyah – say: 

    Two parents referred a dispute concerning a child to one of 
    the judges, who gave him the choice between his parents and he chose his 
    father. His mother said to him: Ask him why he chose his father. So he asked 
    him and he said: My mother sends me every day to the school and the teacher 
    hits me, but my father lets me play with other children. So the judge ruled 
    that he should stay with his mother and said: You have more right to him. 

    Our shaykh said: If one of the parents fails to teach the 
    child and instruct him in what Allah has enjoined upon him, then he is a 
    sinner and he has no right to guardianship over him. The one who does not 
    fulfil the duties of his guardianship has no right to guardianship. He 
    should be dismissed and another person should be given guardianship, who 
    will do what is required, or he should be put with someone else who will do 
    what is required. The aim is obedience to Allah and His Messenger only. 

    And he said: If it so happened that the father marries a 
    woman who does not take care of his daughter’s interests or look after her, 
    and her mother will take better care of her than this wife, then custody in 
    this case should definitely go to the mother. 

    And he said: It should be noted that there was no general 
    text from the Lawgiver giving precedence to either of the parents in 
    absolute terms, and the child should not be given the choice between his 
    parents in all cases. The scholars are unanimously agreed that neither of 
    them should be singled out in all cases; rather one who is an evildoer and 
    heedless should not be given precedence over one who is righteous and of 
    good character. 

    And Allah knows best. 

    Zaad al-Ma‘aad, 5/475, 476 

    Shaykh Muhammad ibn Ibraaheem (may Allah have mercy on him) 
    said: 

    The custody of the young girl goes to her mother so long as 
    she does not get married or until the girl reaches the age of seven years, 
    when custody passes to her father so long as she will not be harmed by 
    staying with her father. 

    In the case of an older girl, custody goes to her father so 
    long as she will not be harmed by staying with her father’s new wife. 

    Fataawa al-Mar’ah al-Muslimah, 
    2/874 

    And Allah knows best.