US Fed hired BlackRock to run a portion of their quantitative easing program
forbes.com
forbes.com
If you want to see another example, during the covid crisis the fed ran a process to decide which custodian to run the Main Street Lending program. They set up the parameters of how it would work and then different custodians could submit proposals ond how they would run it and what their costs would be. This is totally standard for government work of this kind.
https://www.federalreserve.gov/monetarypolicy/mainstreetlend...