You have been given some good advise about the flea bombs but the problem started because you let the cat go outside.
...
In a way that’s correct, but I know cats that stay outside all the time and don’t have fleas. They don’t get any kind of treatment either. The problem is once the fleas get inside they breed like crazy in the carpet and upholstery.
The fleas don’t live on the cats, they suck blood and then get off. If they get off in the house you have a problem.