Lemme see if I can give a cogent description of a situation that I think happens, and could be seen as a fraud.
Any company, even a NON-profit, might hire or contract with a third party for some goods or services, and that third party might be a FOR-profit company. So the money you donate to the NON-profit (or the government grants, or whatever source of capital) might go, in part, to that 3rd party FOR-profit company.
Can you see where this is going? The directors (or other big-wigs) of the non-profit might actually want to make some money. So they set up a third-party for-profit company with themselves (or their buddies) as directors, and contract with that company for some services at mutually agreed upon but exorbitant rates. So they are funneling money away from the non-profit to the for-profit, of which they are also directors (or at least investors). So they and all the investors in the 3rd party company are getting profits that are, in fact, actually defrauding the non-profit company.
This is (supposedly) blatantly illegal, but so shot full of loopholes that I think it happens, with variations, a lot.
Possible case in point: I am a “member” (e.g., customer) of a major NON-profit HMO which owns and operates its own network of hospitals, other facilities, and hires the doctors and other staff. But not really! The company really does only the administration of the HMO, and contracts out to third parties who own and operate all the facilities and hires all the doctors, nurses, lab techs, etc. They do, like, 99% of their business with just ONE such third party, and the general public typically sees the HMO and that 3rd party as one big company with a long name. This 3rd party that actually provides the medical facilities and services is a FOR-profit company (if you look closely enough at the right documents).
So the money you pay the HMO is really getting mostly spent in a FOR-profit company, with (I would assume) only a small portion of that money being spent by the NON-profit company for administration. I think there must be some kind of a racket going on there.