Exterminated is a very strong word! Do they want every single fox in the country to be annihilated (and every fox in a stable loving relationship too?? That is what extermination means.
What would then happen to the corpses of wild animals which had died naturally? Foxes eat more carrion than they do prey caught live and remove these carcasses before they go rotten. They are the binmen of the woods. The live wild animals, like the poor pigeon, take their chances with the fox, as they do with injury, infection, hunger, cars and all the other hazards.
If there are too many foxes in an area to feed themselves on the wildlife and they start on lambs, cats and dogs, then there needs to be a cull - soberly and with care, not as a jolly day out - but if that removes every one and creates a vacancy for a predator, then the post will soon be filled by one of the youngsters old enough to leave a home nearby in the autumn.