Should I report the car stolen?

bad plan, do not make a false report, yes the car is missing, no its not stolen, she gave up the keys. She can however make a police report, and ask the police to help get her car back. Call the police, explain the story and ask them what is the best thing to do. At least this way she can get the car back, but in the future tell her to never let numbnuts take off with the car, The police are not going to do this over, and over, once maybe, best of luck, but do not lie about the issue, that gets your mum in trouble.