Carol didn't do it...It was that crazy ass Lizzy
It was definitely Lizzie.
Carol wasn't gonna let them kick a kid out into the world by herself when they had shelter and perceived peace/stability at the prison. She took the fall for the kid in hopes that Lizzie would be safe.
It's gonna come up though.
I think that Tyrese will grow close to the girls and Carol, to the point that he has a strong connection with them. Then, when they meet up with everyone else, he'll find out, and it will shatter his world, but it won't be like if he somehow found out about it right now.
The only way he finds out right now is if Lizzie or Carol tell him out of nowhere (neither would), or the other girl slips up and says something. Other than that, get ready to watch Tyrese, Carol, and the girls bond until they meet up with everyone else.