If I have a 70 x 300 IS canon lens and add a 2x extender, how does that affect the image stabilization?

I know having more magnification will lessen the effect of the IS. I guess I don't know how the IS inside the lens works. If it's not directly attached to the camera body, how can it get power? I'm on a budget, so is getting the extender better than getting a cheap telephoto lens?