The charge is a racketeering conspiracy – I don’t think it’s dependent on the college’s being victims of a fraud.
The indictment is here: https://int.nyt.com/data/documenthelper/656-college-admissions-and-testing/a01f87291e255a5f7c80/optimized/full.pdf
From paragraph 38:
I think that if there were admission staff members who knew about the racket – then those individuals are co-conspirators. The colleges are separate entities, incorporated as either nonprofit educational institutions operating under a board of directors (for the private colleges), or else as public entities for any public colleges involved. If a staff member at the college – even at the level of a Dean of Admissions or University President – is involved, then that doesn’t change the basic underlying facts.-- then those staff members are committing crimes against the entities that employ them – especially if any of them took bribes. It wouldn’t vitiate the criminal charge if it turned out that the Dean of Admissions at one school or another was involved – it would just add another layer to the depth of the conspiracy.