My Nikon 500mm f/4 AF-S D IF II lens won't auto focus on a Nikon D500 until I attach a Kenko 1.4x teleconverter — then it autofocuses fine. Does anyone know what's going on?
The electronics guy in me would put some money on some bad contact somewhere on the lens, that the Kenko converter mount exercises a bit differently from the camera one. Some contact cleaning fluid can work wonders. Make sure you don't spill any on the lens or inside the camera.
It could be a bad connection in the contact pins, or it could be a similar connection problem with the lens that is more internal in nature.
Many third party teleconverters report 'false' lens information to the camera in order to try and get around requirements regarding a TC+lens combo's maximum aperture for the camera's AF system to even attempt to focus. If your TC is reporting a spoofed lens ID to the camera, rather than passing through the lens ID from the actual lens, it could be allowing you to shoot with a lens that itself is not, for whatever reason, properly providing a lens ID.
The obvious first step is to try cleaning the contacts and checking the spring loaded contacts (Nikon F mount puts the springs on the lens side, rather than on the camera side like many other makers do). If that doesn't help, you might try the lens on another Nikon body.