Child Custody in Sharia Law

My friend, a woman who is an American Citizen, has converted from being Jewish to being Muslim. This is because her boyfriend's father, although an American Citizen himself, is from Pakistan. Her boyfriend follows the Koran very strictly and about 3 months ago, he had arranged a Nikkah for them.

They did not get married in the American courts.

My friend's "husband" (under Islamic law) wants them to move to Dubai, where his father and sister are now living, but a few weeks ago, she discovered that she was pregnant.

My question is regarding the consequences of going with her "husband" to Dubai, with the baby, then deciding that she wants to return to the United States. This could be either because she is too homesick (she is very close to her parents, grandparents and extended family here in the U.S.) or she is suffering from culture shock or she just does not feel comfortable as a woman in Dubai.

Now, let's say that her husband through Muslim law only, and the father of her child does not want to return to the U.S.

Can my friend legally return to the United States with her baby? Would she need written permission from her Muslim husband to board a plane with her child? Is it possible that the only way for her to return to the U.S. is without her child? At the very least, could this become a legal nightmare that would be prohibitive for her to return to the U.S. with her baby?