We asked our accountants about this a while ago and as I recall, what the VAT person said was this. If a VAT registered entity (a flying school), is acting only to collect the fee for someone who isn't VAT registered, (such as a self-employed FI/FE) and they don't add any further charge, or take a cut, but pass on the whole amount, then it can be processed as a zero VAT transaction.
I can't back that up with an extract from a HMRC publication or anything, just what we were told.