"After a number of hours of walking, day began to dawn and they eventually found someone who helped to make a phone call for help," a group spokesman said.
The authorities believe that criminals confused the students with rival gang members.
Witnesses reported seeing the victims being bundled into an unmarked van.
It took months before the boy learned enough English to tell his new American parents he wasn't an orphan, but by then it was far too late.
Asaib Ahl al-Haq, Kata'ib Hezbollah and the Badr Organization are the principle focus of the investigation.
U.S. sources said Washington had no reason to believe Tehran was involved in the kidnapping.
Analysts warn that as the group lose control of oil fields, they'll make money in other ways.
The 16-year-old was seen being bundled into a van, witnesses said.