There are very few zoom lenses I will consider using an extender with -the exception being the 70-200mm II (and III). With Canon prime L lenses, I am often comfortable using the 1.4x III. - an extender I will often use with the Canon 400mm DO II. The 2X III extender however is something I usually try to avoid - there is a large drop in image quality and sharpness. There are perhaps two lenses I will make an exception with the 300mm f/2.8 II and also the 600mm f/4 II. The starting point for these two lenses is exceptional and both , in my view, produce very acceptable results with the 2x III.
photo with 2x III on 600mm f/4 II