terminology for Islamofacists
Despite the all-knowing derision of the term Islamist, Islamist is the usual term in the Islamic world for these people. It's not a Western simplification of Islamofascist (Christopher Hitchens' coined term) or Islamic fundamentalist. In a formal setting it's the most correct term: a political ideology centered around Islamic law, and especially recently with aspirations toward restoring the Caliphate and uniting all Muslims.
You also hear the term jihadist in places like Pakistan where there is an entrenched secular culture, but it's not universally popular.