There's no fee to pay unless there is a change in trusteeship of the group, which means a change to the C of R.
It would be normal to receive a receipt for your share in the group's assets and liabilities rather than just a share in the aeroplane, and it's up to you to determine the seller's title before buying.
As has been said, the trustees are obliged to update the share ownership grid, but that information is not published and doesn't appear on the C of R.