Sigma guys, help me decide!!

I have a big dilemma on my hands... I am looking at getting a new lens but I really need 2..... I want to be able to take pictures of my little one at gymnastics indoors so the Sigma 70-200 2.8 is on the top of my list. BUT... I also enjoy taking pictures of wildlife. I have a 75-300 now (that's why I am upgrading) and I can live with the 300 but would like more range so I am also looking at the 50-500 also. I have a 50mm 1.8 so I can take the pictures indoors, just have to be close, usually closer than they like me to be!! So I guess the real question is how well does the 70-200 2.8 work with the 1.4X or 2.0X extenders? Does it still autofocus? Light wouldn't be an issue because I would be outside when using the extenders so my only issue is picture quality and autofocus.

Thanks

I think...

If I remember correctly it will still autofocus if you stay
above f5.6.... Someone will correct me if I'm wrong. So if
light isn't the problem then they should work. The 2x looses
2 f-stops so the light to ISO will need to be around f-11 on the
subject to keep the lense/2x at 5.6.....

I have the lense and the 1.4 & 2x and they work perfectly,
sometimes though I have to manual focus in low light.

Good luck, Rod