Ah. It’s a good question, certainly. From your reply here-

Jihad here is used correctly. I too am not a fan of the suggestion that “Jihad” and “holy-war” are synonymous if that’s what you’re getting at. But while non-violent connotations of the word (such as striving towards good, struggling to maintain faith, advancing the status of the Muslim community, etc.) are important, Jihad as a term for warfare in the name of Islam, both defensive and offensive, are undeniably an accurate useage of the term as well. This is just one kind of Jihad, and it’s usually called the “Lesser Jihad” as opposed to its far more significant spiritual counterpart.

Actually, jihad in war (Jihad bil saif) is only a defensive war. From here- “Jihad bil saif is only a defensive war. If an offensive war is waged, or if one attacks without being attacked first, that is not a form of jihad. It’s just war.”

And if we’re arguing that extremist groups are trying to defend their country- which is true, that is their motivation- you could say that it’s a defensive jihad. But extremism in any form is forbidden in Islam, and the work of extremists isn’t really founded in Islam. So by perpetuating a wholly wrong ideology that isn’t supported by Islam, they’re not really defending themselves or Islam.