There is a certain amount of thread creep here. The original question involved a PPL training flight from a temporarily unlicensed airfield. The correct answer to that question is that a PPL training flight must not take off or land anywhere other than a licensed or government airfield.
If transit from the unlicensed airfield to the licensed airfield is required before starting or after finishing the PPL training (or any other training or testing for a licence or aircraft rating) it must be conducted as a private flight, which means that no valuable consideration may be given for it. (i.e. the student may not legally be charged for the flight time in transit).
Then the creep started -
homeguard suggested that
....if you are acting outside of an AOC you must be exploiting the exemption given to training within a club
Let's assume that we are still talking about training for a licence or aircraft rating. There is no requirement for an AOC to be held in order to conduct flight training, such a certificate is required only for public transport. There are plenty of organisations out there training for licences and aircraft ratings without holding an AOC. Similarly there are plenty of organisations providing training without being a 'club', the TRTOs for which I freelance, for example, or OAT, FTE and other major training organisations.
The goalposts then shifted even further when
llanfairpg asked
Are you saying that all the freight carriers i have flown for then didn't need an AOC after all?
I have to say that I wasn't too sure which bit of the quote he was referring to but he seemed to suggest that carrying freight was not aerial work. In any event, the AOC in that case was clearly required for the public transport aspects of the flight, the fact that training was taking place being totally irrelevant (as Whopity has pointed out elsewhere). Of course, llanfairpg's example is even less relevant since whatever training he was providing on his public transport flights was obviously not training for the issue of a licence or aircraft rating.
llanfairpg went on to state
...it is my understanding that if you and the student are not members of a club and you take them flying for the purposes of letting the student handle the controls it is public transport and therefore you need an AOC
If this were true then every training flight conducted by OAT, FTE, etc., which are not 'clubs' and do not hold AOCs, would be illegal public transport - I think not!
To summarise -
Flights for the purpose of training or testing for the issue of a licence or aircraft rating may not take off or land other than at an airfield that is licensed for the purpose or is a government airfield.
An AOC is not required in order to provide training or testing for a licence or aircraft rating.
Training for a licence or aircraft rating does not have to take place within a 'club'.
The confusion in the latter case may be related to the the privileges of an FI who holds only a PPL. The privileges of a PPL do not include aerial work except for glider towing, para dropping or flight instruction and testing
"in an aeroplane owned, or operated under arrangements entered into, by a flying club of which the person giving the instruction or conducting the test and the person receiving the instruction or undergoing the test are both members" Note that it is the qualification of the FI that requires the club environment, not the flight instruction.