what’s the protocol for injured stray animals?

I found an injured cat on the road and not sure what to do with her. In the US in this situation, I can bring the cat to a vet/shelter and they will take them in and take care of them. I immediately took the cat to a vet and they basically just gave her fluids and sent me home with her. She was bleeding and had a fractured jaw and when I asked what I should do with her from here, they just told me to find someone to take her that can give her regular vet care or take her in myself.

So now I have this cat in my possession and I’m unsure how to proceed. Never mind my apartment not allowing pets, I travel a lot and could probably not be able to keep a pet. If I set her free outside she will probably die. What options do I have? I live in Nagasaki prefecture, and as far as I can find online, there aren’t any animal shelters I can give her to. How can I help her?