What is a right-wing extremist and are they necessarily racist?
A right-wing extremist may be described as someone who largely disagrees with everything a left-wing extremist believes. And vice versa. I realize this is a vague definition but the problem is that there is a right/left continuum and depending where you are on that continuum you perceive those closest to your position as pretty reasonable and those farthest from you as zealots or dreamers or just plain dangerous. As for racism it would be dishonest to say that many racists don’t vote Republican. But that doesn’t mean that racism is part of the party platform. Many criminals vote for Democrats but that doesn’t make Democrats pro-crime. One certainly does not have to be racist to be a right-wing extremist.
It’s very hard to answer because the terms ‘right/left wing’ and ‘extremist’ are highly subjective and subject to anachronistic interpretation. To answer your question from a modern, British view, I believe a right wing extremist would be one that believed in a near totalitarian government, enforcement of religious laws, aggressive military actions and basically very little support for the welfare state or its traditional users. They are not necessarily racist, though often right wing extremists do have a blend of overbearing ethnic nationalism that can result in resentment of other races.
