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Is it possible to detect the permissions of any running process in the background. E.g. if there is an audio recording app running and it has access to the microphone?
If you mean detecting the permissions of other apps, then no, it is not possible using any documented API. Apps are strictly sandboxed and can see almost nothing about each other, aside from the name of the the other sandbox folders and, in some cases, the public icon file. You can't even get any sort of listing of which other processes are running.