Just miles from where former Guantanamo Bay terror suspects have resettled, American warplanes take off from Qatar’s al-Udeid air base in the global war on extremism.

The contrast in images illustrates why tiny but rich Qatar is an intriguing player in what President Barack Obama says will be a long battle to stop and eventually destroy the Islamic State group in Iraq and Syria.

Qatar plays an outsize role as a U.S. military partner. It gained public praise from Obama for brokering the controversial deal that freed Army Sgt. Bowe Bergdahl from Taliban captivity in May in exchange for the release of five senior Taliban officials who had been imprisoned for years at the U.S.-run Guantanamo Bay prison in Cuba. Qatar promised Obama it would keep the five under watch for one year, although they would then be free to leave. The Obama administration also praised Qatar for its role in securing the release of extremist hostage Peter Theo Curtis. […]