1-What is the meter prism on your F2?
If there is an "A" on the front of the prism, then it is an AI body and the lens will need to an Ai type or having been converted.The view finder is the one that originally came with the camera, split screen I think> It's the Photomic with type A focusing screen.
2-You said your 50mm lens is a Nikkor-H. That seems to be a Non-Ai lens. Look at the following image of an Ai version:
You can clearly see the silver metal "rabbit ears" for metering with Non-Ai bodies. Look carefully to the second row of apertures in small letters. Can you see in front of F11 a black plastic ridge starting there? That's the Ai coupler.
For comparison, here is the Non-Ai version:
3- If your Nikkor 50mm doesn't have the Ai coupler, it isn't coupling with the meter and that's why you get it to meter inside as the camera will see it at F2.0, but it is showing overexposure outdoors as it still sees it at F2.0, regardless if you change apertures. I bet the Soligor is an Ai lens and that's why it is working properly.
You can still meter with the 50mm, but it has to be at working aperture. for that just use the Deph-of-field lever to close the lens and to meter.