I think the many of the ones that represent radical social change or a vector away from social convention do, and their extremism is often in direct proportion to the societal resistance they meet up against.
There is the vanguard, or the initial wedge. They are often made up of purists, uncompromising true believers, people that are energized through anger born of perceived injustice and a desire for change so strong that they are willing to put themselves forward and suffer the slings and arrows of resistance.
Then, as the movement matures, there is the core that feel betrayed by the moderation of the late comers, the or what becomes the more compromising moderate majority of the movement. The core minority are often the same people as the vanguard found in the first stages of the movement’s evolution and they feel compromising will pervert and soften the original goals (wise opposition can often exploit this stage to divide, confuse and weaken the movement). At the same time, they sometimes believe change will occur faster if more forceful tactics are used.
Society’s reaction to extremists is almost invariably hardened resistance. Society, by nature, is primarily interested the modicum of stability required to raise it’s children, the biological imperitive. Radical change, whether justified or not, threatens this interest. Extremist proponents within a radical movement will cause society to focus on protecting itself from this threat and not focus on any possible rationalizations that could allow adaptation to the desired change.
Does every group have a extremist element? I really don’t know if all of them do. But there are enough violent extremists to cause society to be well on their way to develop a whole school of thought on the subject.