For this to unlock you need to PURCHASE one of every Group A car.
That's a total of 4 cars:
1) Subaru Impreza 1995
2) Lancia Delta HF Integrale
3) Escort RS Cosworth
4) Mitsubishi Lancer Evolution VI
These will randomly become available in the "Purchase Vehicles" section of your garage, usually 2 at a time. Depending on the condition Group A cars will set you back between 245-290K credits. Once the selection of cars is settled you will need to wait for a set amount of time for the selection to reset so you can get new cars.
IMPORTANTLY you only need to PURCHASE every Group A car to unlock this achievement. You DO NOT need to actually have all 4 cars in your garage at the same time. You can sell them almost as soon as you buy them (at a loss of about 50-90K credits) - just make sure the achievement tracked your progress. Also, if you got the Subaru because you got the Deluxe edition it does NOT count. You still need to buy a 1995 Subaru to get this achievement.