Buying an Airline Ticket for Dual Citizen

I have a dual citizen family member, American and Thai, who came to Thailand at the beginning of COVID. I need to get them back to the U.S. The first names are different on both passports. So, I am aware to use the Thai passport leaving Thailand, and the U.S. passport to enter the U.S. My question is which name and passport do I use for the ticket? And follow-up question, does Thai immigration only check the passport or the passport and the airline ticket? Any help is much appreciated. Thank you.

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