If you think the appeal was conducted improperly (not just that didn't go your way), you can take it to the ombudsman.
The following is not reflecting on Bouga's case.
Last year, two of our failed appeals were taken by the parents to the Ombudsman. In both cases, our clerk got a grilling (in the school holidays!), but as everything had been correctly carried out in the appeal, both were rejected by the Ombudsman. From what I heard, the parental case was "They turned our appeal down. We'll go over their heads to the Top Person, shout a bit, and get their decision overturned".
There are a few appeals that are not carried out correctly, and that is what the Ombudsman is for. But as WP says, it's not just that the panel found against you - something must have been done wrong. Such as a the school side being seen to slip the panel members a bundle of bank notes, the chair of the panel being a governor of the school in charge of admissions. That kind of thing.
Most of us do our absolute best to be fair to all involved (including the oft-forgotten children who passed with a high enough mark and will be joining the school in September). So please, people, only take it to the Ombudsman in case of a genuine problem.
The panels decision is final!