If you select a parent and a child--in your case, a group and the skeleton inside the group--the transformations you make stack. So as an example, if the child moves 1 unit and the parent moves 1 unit, the total distance covered by the child is 2 units. If you want something to move as a group, you only move the highest…
It looks like you're mixing usage of groups used for animation and groups for organization. For instance, I could have a group called Spheres_GRP, and have redSphere_GEO, blueSphere_GEO, and greenSphere_GEO in it for organizational purposes. I could also group together redSphere_Outter_GEO and redSphere_Inner_GEO, and…