Key fobs don't work properly on a 2006 Bentley Continental Flying Spur

[desiled]

I just bought a 2006 Bentley Continental Flying Spur 3 days ago and I'm having issues with the key fobs.
The fobs won't open the doors nor unlock the trunk. However, once I start the car with one of the fobs, then after cutting it off, the fob will lock or unlock the doors and release the trunk for about 5 minutes at the most. Then, it won't unlock the doors nor the trunk again.
Also, according to the manual, with the fob, I should be able to lift the door handle and unlock and enter the car. That won't work.
It should also be noted that with the car off, the door lock / unlock button and trunk release button inside the car doesn't work. However, with the car running, the inside door lock / unlock button and trunk release button will work.
I haven't changed the key fob batteries yet, but the previous owners were having the same issues.
Any ideas?