As of now, the user has the ability to grab the origin of a selected group and manually change its location. However, there isn't a way to do this through the API.
One of the biggest issues with this is when you create a group within the API, especially when trying to base it off of another group. Right now I have to have a separate section of code depending on whether the group has already been created just to ensure the group is placed in the right position when it would be so much easier to adjust the origin.
This idea has already been suggested here a while back, but didn't get enough traction at the time. In my research I've come across many people who were struggling with ungrouping and regrouping elements in the API and they are all likely running into this same issue.
