Google's process produced a price of IIRC $85/share and on first-day trading soared to >$100 so there was a lot of discussion at the time about how efficient that process was.

But really that's how all IPOs work, basically. You have one of more investment banks that underwrite the offering. They're basically guaranteeing to sell a certain number of shares to their clients at a given price. Those clients can be institutional investors, pension funds, high net worth individuals and so on. But there's a feedback loop here where clients might push back on a certain price.

IPOs love these sorts of investors because they tend to buy and not sell. If everyone sells the IPO will flop. Retail investors are far less "loyal".

The IPOing company also has levers where they can manipulate the price, most notably on the supply side ie by limiting or expanding the size of the float. SpaceX's float (as a percentage of the company) was actually really small.

What's unique about the SpaceX IPO was that it would immediately become one of the world's most valuable listed companies so there'd be a lot of induced demand from index funds. The underhanded (IMHO) aspect to all this was that the rules were deliberately changed so passive investors would be exposed almost immediately rather than first allowing some form of price discovery by the market. NASDAQ capitualted. S&P did not.

I guess the real manipulation here is the fiction that SpaceX is an AI company, which ultimately goes back to a series of bailouts for terrible decisions going back to the Twitter purchase. SpaceX's AI pitch was orbital data centers, which make no sense, and using their ill-gotten NVidia chip allocations to rent them to Google.