I wonder why, in view of the seriousness of the accusation, the CAA didn't lift the phone to call the examiner to enquire about why he (according to their info) did a test when not authorised.
It would have found the solution long before their letter was ever written.
Indeed, assuming that the op contacted their school, the examiners reputation might have been quite damaged by this little incident, and the CAA are the only ones at fault here.