What should I do about my friend's cat?

My friend has a cat who is ALWAYS alone. She goes out of town for weeks at a time and then comes home for a day or two and then leaves again. She is battling alcoholism and she can't take very good care of her home, herself, or her cat. The cat always has food, and usually fresh water, but his litter box is always dirty and her place is filthy. I can't take the cat and I don't know anyone else who can. I think that he may have some health problems. He still seems like a very happy cat whenever I see him. He is allowed to go outside sometimes and he never runs away. He's lived in 3 or 4 different homes. I just feel so bad for this cat, and I know shelters are overcrowded. What should I do about him. Is it wrong for me to take him to be put to sleep? Wouldn't it be better then to take him to an overcrowded shelter where he is locked in a cage and then possibley put to sleep? Or should I let him live his lonely life in squalor? I've talked to my friend about this multiple times and she is lost. Everyone tells her that she has to get rid of the cat, and she goes back and forth about the issue. She loves him a lot, but can't even take care of herself right now. Any advice, no matter how harsh is welcome. I'm really stressed out over what I should do for "Georgie"!