Grinding noise after startup

Whenever I start the car, which gives me no trouble, there is immediately a grinding noise for about 2 seconds, and then it stops. It's an awful noise, sounds like gear teeth that are not aligned, and definitely isn't okay. To be clear, it doesn't happen before, or during ignition, only right after the car has started. Then it stops. No other problems with engine sounds. Also, it will not do it if I have been running the car for a while, and the engine is hot, and turn off and start again. Only when I start the car from rest. I'm told there is a strong exhaust smell right after I start my car as well, if that has anything to do with it.