Sorry, the description is still kinda vague and I still can't see your images. But here's what I think you're asking....
1. By 'still' do you mean when you move the group, you want the contents remain in their current
global position...like they're not part of the group?
If so, I need to know why you'd want to do this. Normally, if you don't want something to move with a Group, you would remove it from the group. But if you're trying to hold the position of the contents while changing the origin of group, you probably just want to use the Set Origin tool. But if you really want to move the group (not the origin) while preserving the global position of of its contents, maybe one of those scripts I mentioned earlier would help.
(I just saw Wes' response...or that.)
2. Or do you mean that when you move the group, the contents appear to move independently of the Group's position and you wish to prevent that?
I'm going to guess you mean this one since it's a more common situation. In this case, I wonder if your content layer is offset in Z-space? This can make the child layer appear to drift when you move the parent group layer because of parallax. To fix this, make sure the Z-position is set to 0 (to align with the group's position.)
BTW, you can visually check for this problem by using the Orbit tool and looking at your setup from another angle.
Hope this helps. If not, I think I need to see the images or a clearer description of the problem.