FrontPage Magazine has an article here elaborating on why ISIS believes it represents true Islam.
ISIS is taking umbrage at critics who challenge its claim to represent true Islam. In the latest issue of its magazine Dabiq, published on July 31, the author of an editorial entitled “Why We Hate You and Why We Fight You” repudiated “the Islam-is-a-peaceful-religion crowd.” That crowd would include President Obama, Hillary Clinton, Pope Francis and other world leaders, politicians and opinion makers who have pushed the narrative that ISIS is a deviant cult made up of violent extremists whose acts of terrorism have nothing to do with Islam. To the contrary, claimed the author of the Dabiq piece, “jihad and the laws of the Shari’ah – as well as everything else deemed taboo by the Islam-is-a-peaceful-religion crowd – are in fact completely Islamic.”
... ISIS’s literal reading of sharia law, which is based on the Koran and the sayings and actions of Prophet Muhammad, is closer to traditional Islam than the fanciful version of “Islam-is-a-peaceful-religion,” propagated by Obama, Pope Francis and the Islamists’ other apologists.
... Ironically, it was President Obama, who at the outset of his first term in effect legitimized the brand of traditional Islam on which ISIS’s ideology is based, whether he meant to or not. Saudi Arabia is home to Islam’s two holiest sites in Mecca and Medina. It is also home to Wahhabism, the supremacist Islamic ideology based on a literalist reading of sharia law that ISIS shares. During a visit to Saudi Arabia, on the eve of his speech to the Muslim world in Cairo, Egypt in June 2009, Obama told reporters "I thought it was very important to come to the place where Islam began.” Islam began the same way ISIS is carrying out its literalist dogma today – a religion built on jihadist conquests to subdue disbelievers and make Allah’s rule supreme in the world.
When President Obama delivered his June 2009 speech to the Muslim world, hosted by Al-Azhar University and the University of Cairo, he went out of his way to praise Al-Azhar University. He called Al-Azhar University a “beacon of Islamic learning,” a place “that carried the light of learning through so many centuries, paving the way for Europe's Renaissance and Enlightenment.”
This “beacon of Islamic learning” decided it was inappropriate to condemn ISIS as un-Islamic .....
... In short, ISIS in a very real sense represents Islam in its original literal form, using a blend of archaic barbarism and 21st century technology to continue Prophet Muhammad’s legacy as jihad.warrior.