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However, we don't know the specific situation here. Is this just a "general" question or is OP talking about a specific situation such as in a divorce?
Child custody rights for mothers differ from situation to situation and also from jurisdiction to jurisdiction. It must be noted that the rights of custody are in the favor of the mother, most times, especially if the behavior of the father is abusive or tending towards domestic violence.
I guess the OP will come back & tell us if this is in regard to a specific situation.