In the UK, Privileges are listed in Section XII of the Licence.
Is this true EASA wide?
For an FI with CPL knowledge: "FCL.905 FI applies as in /(a)/(b)"
Where (a) =
a PPL, SPL, BPL and LAPL in the appropriate aircraft category;
and (b) =
class and type ratings for single-pilot, single-engine aircraft, except for single-pilot high performance complex aeroplanes; class and group extensions for balloons and class extensions for sailplanes;
Checking the Licence would seem the easiest way to confirm what an FI is entitled to teach.
I would assume that for an FI without CPL knowledge the first entry would be "/(a) LAPL only"; with the BIG question ?? being whether there is second "/(b)" entry.
Of course this doesn't help if someone is planning an FI course and wants to know beforehand which Privileges their Competent Authority will actually enter on their Licence following successful completion
Can FIs with a PPL instruct the initial stages of the integrated courses?
Unless I have misunderstood, if they can teach for the PPL, then Yes:
FCL.910.FI FI — Restricted privileges
(2) in all integrated courses at PPL level, in case of aeroplanes and helicopters;