B767 Captain reported that the aircraft APU failed three times to assume the electrical load after engine shut down. During a repositioning flight; the aircraft lost navigation and communication systems after which the Captain elected to perform an air turn back to the departure airport.

Narrative

We started this maintenance ferry from ZZZ1 to ZZZ with an APU that was not verified for ETOPS flights so we had to run the APU for the entire flight. Upon parking in ZZZ we lost the APU generator after engine shutdown. The APU was still running but the generator could not pick up the electrical load and the airplane lost all electrical power. We made a maintenance write up and went to hotel.The next day before our scheduled flight from ZZZ to ZZZ2 we told Maintenance that we were going to shut the engines down after start to verify that the APU generator can handle the load in case of engine failure or failures in flight. The APU generator was able to pick up the load and we proceeded on our flight to ZZZ2. Upon parking in ZZZ2 and shutting down both engines we lost the APU generator once again and the airplane lost all electrical power. We made another maintenance write up.During the day I called ZZZ2 Maintenance to find out what the status of our APU was and to let them know that we wanted to do another engine run before our flight to ZZZ3 and test the capabilities of the APU generator. I was told that it would be no problem and was welcomed as ZZZ2 does not have qualified engine run mechanics. We went to the airport early and received authorization from Maintenance Control to perform the engine run. We started the engines and then with electrical and bleed air load on the APU we shut down the engines and the APU generator once again handled the load.We took off headed to ZZZ3. We were at our cruising altitude of 40;000 feet and about 250 nm from our CP we lost the right FMC. We accomplished the non-normal checklist and also performed a Navigation Accuracy test due to our concern of headed to [foreign country] airspace. During our accuracy test we discovered the primary IRS was displaying 6.72nm and our RNP for the airspace was 4.0nm. We tried to establish contact with Dispatch multiple times for assistance and coordination but were unsuccessful either via Satcom or HF radios. Due to our concerns about Country X airspace being controlled by the military; our concerns about our navigation capabilities and with no way to contact dispatch directly we felt as a crew that the safest course of action was to return to ZZZ2.Also; the fact that we had very little confidence in our APU and we were down to 1 autopilot since the center autopilot was deferred and with the loss of right FMC we lost the right autopilot plus our inability to communicate with dispatch headed to a non English speaking and military controlled airspace; all contributed to our decision to return to ZZZ2.We were able to speak to Aircraft Y that was also headed to ZZZ3 and they let Dispatch know our concerns and our intentions to return to ZZZ2. We were finally able to contact ZZZ radio and received a clearance to return direct to ZZZ2. We did not declare an emergency.Upon shutting down our engines in ZZZ2 the APU generator for the 3rd time could not pick up the load and the airplane lost all electrical power. We wrote it up again.