Heaven is a radiator in cold weather

This is what I call my 'stranger cat'. He half lives on my property, but he's quite a Tom cat and feeds around various houses, but very sweet.

He was definitely owned as his collar was being replaced when he lost one, but I'm not so sure now. It's hard to tell as he has always been a real 'Tom', but it does look like he is definitely living outside at present. His fur is very thick, and there are a few places he can sleep (including a cat house under my porch which is very sheltered and warm), and I do put food out for him.

I let him in for a couple of hours today and he just sat under the radiator.... but I can't let him stay as he sprays and pees all over (and I can't keep cleaning cat pee from the carpet). He seems to be ok, and he has never seemed to be an indoor cat, so hopefully he will be fine if I keep an eye on him. But I do feel so guilty when I turf him out :( I will see about calling in the cat charities, but I suspect that he just isn't a house-cat and will only stray again.