DaveReidUK:
No, the NTSB will carry out the investigation, after (or during) which they will make any Safety Recommendations that they consider appropriate, directed to the airline, manufacturer, regulator, etc.
The FAA is involved in this one, too. But, for flight crew conduct of the flight; i.e., an enforcement investigation. Because it is a foreign crew if the FAA determines there was regulatory non-compliance they will forward it to the State Department for a diplomatic resolution.