There's no need for the cat to go outdoors. You're the person. Don't let the cat out. If you have to post a sign at every door, so other members of the household don't let the cat out, post it. If you have to play "dodge the cat" when you bring in groceries, do it.
Hopefully, you have a screened porch or sun-room where the cat can get some fresh air.
What does your son want for the cat? When he takes it back, will it become an indoor cat again? Better to just keep it in.
And everything Greg B said about disease and injury and fleas & parasites and vet bills... you'll avoid all of that stuff if you keep the cat in.
Play with the cat & give it extra cuddling, and it'll be fine.