The Associated Press refers to it as the Islamic State group - to distinguish it from an internationally recognized state - or IS for short, usually as an adjective before the words group, organization or extremists.
What is the difference between ISIS and ISIL? Here's a brief summary from the AP:
The group traces its roots back to Al-Qaida in Iraq, which declared an Islamic State of Iraq (ISI) in 2006. The name never really caught on, however, because the militants were never able to seize and hold significant territory. That began to change when the group expanded into neighboring Syria, exploiting the chaos of its civil war. In 2013 the leader of the Islamic State of Iraq, Abu Bakr al-Baghdadi, renamed it the Islamic State of Iraq and al-Sham, signaling its emergence as a transnational force while sowing the first seeds of confusion over what to call it. Al-Sham is an archaic word for a vaguely defined territory that includes what is now Syria, Lebanon, Israel, the Palestinian territories and Jordan. It is most often translated as either Syria — in the sense of a greater Syria that no longer exists — or as the Levant, the closest English term for the territory it describes. In English, the group's name was translated variously as the Islamic State of Iraq and al-Sham (ISIS), the Islamic State of Iraq and Syria (also ISIS), or the Islamic State of Iraq and the Levant (ISIL), the term usually used by the U.S. government and various U.N. agencies.
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