First off, the terms 'sharp' and '2x extender' don't work well together...


But yes, stopping down a little does improve sharpness, as you can see in this ISO 12233 comparison. I would suspect that if you go even narrower, diffraction will kick in and sharpness will start to go down again.


As a side note, you don't 'set the aperture to f8 to become f16' - the lens reports aperture to the body, and with the 2x extender the camera 'knows' the widest is f/11, so that's as wide as you can set it.