I'm not sure I understand what you want to do.  You say you can move the hole, but you are not able to change the axis of the hole to a different axis system.

The hole has a centerline axis. Is that the axis you mean?  That axis moves with the hole when the hole moves.  What is the different axis system?

Do you mean when the Move tool is active, and there is a Triad.  The triad origin can be moved by picking on its origin. 

Please explain it with a video or some images that have been labeled for the things you are talking about.