The SEP CRI (or PFA Coach) who is not an FI could not send a "student" solo as they are not authorised to do ab-initio training i.e. their "client" must already hold a valid licence with medical and they are simply providing dual differences training or refresher training to qualified pilots.
To use an example described previously - IF a PFA coach (a CRI) is training a licence holder towards the tailwheel differences requriement, they can not send them off in the aircraft solo until they have certified completion of the diffrences training in the logbook. THis is because they are only authorised to instruct, they are not authorised to supervise solo flights.
Thus an FI could send the guy solo half way through but a CRI can not.
3 Point,
If the pilot held a valid SEP rating and was merely doing differences training then the training could be done at any field. However, sending the guy off in the taildragger without having completed the differences training (i.e. he is not dual and he is not flying within the privileges of his licence / ratings) makes it a student solo flight and thus must be from a licensed airfield.
Differences dual training, IMC rating training etc can all be done from unlicensed strips but there is no student solo element in that training.
As for the student solo while awaiting licence - must most definitely be under the same requirements at all the training prior to test pass. You can not pick hours in the logbook that you wish to "ignore". To go down that raod would lead to certain organisations authorising solo student flights at unlicensed airfields and simply "not claiming those hours" on the application. Not allowed I am afraid.
Regards,
DFC