In France registered car as German driving in Germany “?”

Hello Together , summer is at the door and my mother’s friend has a Caprio that I could drive the summer over, now my concern: The friend of my mother lives in France and is French but also has a German passport, my mother and I are Germans and both live in Germany as we suspect that I can’t just drive the car I wanted to ask how we registered the car so that I am insured on the car. My plan was: 1. The car was my He officially give mother 2. My mother to register the car from 03-10/15 3. Get TÜV confirmed 4. Insure car 5. Enjoy the sun Are there tricks & tips on how to make it easier / or where we can save money? Many thanks to every answer and help already in advance !