How can an unmarried mother obtain authorization to take her young son overseas if she and the child's father have never been married or established custody and the father will not sign the forms?

If the father is named on the birth certificate then the mother will have to file a writ of habeas corpus to request sole custody of the child. The father will be notified of the action and will be given a chance to contest it. The court will then rule as to which parent should retain custody or if it should be shared and if the child can be taken outside of the country. If father is not named on the child's birth certificate all the mother must do is present the birth cerificate at the time she applies for a passport for the child.