You can add a student to a group in two ways:
Navigate to the page of the student. Below the heading ‘Groups’ click on the button ‘Add to group’. Select the desired group and click ‘add’.
Navigate to the page of the group the student is currently in. Drag the student from the table (keep the mouse button pressed) to the new group in the submenu on the left. Release the mouse button above this group.
To move a student to another group, you first have to add the student to the other group. Then delete the student from the curent group.