Suppose an American citizen male went to a foreign country and got a women there pregnant and a baby was born. Two questions: 1. Does the baby have American citizenship? 2. Is the mother guaranteed a green card?
This American male needs to work with an experienced attorney to file an N-600 Application for Certificate of Citizenship. There are certain requirements that have to be met and the immigration attorney would be able to analyze his situation. Based on the facts you mentioned the mother has no basis for a green card. She is eligible for a green card if she gets married to the United States citizen male. Although her son may be a US citizen, this does not entitle her to a green card. A US citizen child cannot file for a parent's green card until the child is 21 years or older. This U.S. citizen male should speak with an immigration attorney to discuss how to satisfy the requirements for filing the N-600 for the baby.
A wife married to a US citizen has the possibility of getting a green card but that is not a guarantee. A child can apply for citizenship.
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it is possible the child is a US citizen, assuming certain other criteria exist. The mother of the child in the foreign country should check with the American Consulate there for steps required to be taken to report the child's birth, under a procedure called Consular Report of Birth Abroad (CRBA). The father has to be involved in the process by signing certain forms. I say it is "possible" that the child is a US citizen because there are other requirements and your fact pattern did not cover these other requirements.
The CRBA, if the child qualifies, becomes primary evidence of the child's citizenship. The mother can also apply for a US passport for the child, which, when approved, is also primary evidence of citizenship.
Re green card for mother, based on your facts I do not see a qualifying relationship making her eligible for a green card, much less a guarantee of such status.