The way you have it mapped, the User is the managing side of the relationship, therefore it will be responsible for updating the join table.
Change the JoinTable mapping from User to Group, and make the groupList property of User so it has the mappedBy attribute. That will change the group to the managing side of the relationship, and make persist/update calls to the group manage the join table.
But take note of that, you won’t be able to simply add a group to a user, save the user, and continue, you’ll instead have to add a user to the group and save the group to see the changes, but having the bi-directional many-to-many hopefully you’ll be closely managing that relationship anyway.