Major flaw in your argument there is that some of these Islamist extremist's you talk about are as British as you or I. Thats like saying a parent is an extreme homophobic and one of the kids turns out gay, would you say its right to kick the kid out?
In reality we do not have the thought police (yet) and if people want to think in this fashion then they can, it is only once they consipre to do something illegal or in fact commit a crime that you can punish them. That is fundamental in a free society, we are free to commit crime (no matter what it is) but once we step over that mark then there will be consequences.
Once you start kicking people out or imprisoning people on the basis of belief or race then we are voluntarily giving up our own freedom (such as it is..) to become a facist police state.
Personaly i do not think there really is an immigrant problem at all and it is simplya convinent smoke screen for the govt to wave in front of people while they further thier own agenda.