At this time, it is not possible to change the owner of a group.
Changing the owner of a group from one user to another would also require changing the owner of the contacts within that group. This cannot be done as the contacts might already exist in the account of the other user.
As an alternative you can export the contacts from the group and then import them into the account intended to be the new owner. If you choose to do this, any changes that are made to the contacts in the original owner's account will not be reflected in the new owner's account, so you may want to consider deleting the group from the original owner's account if that is no longer being maintained.
For information on shared groups, please see the article below: