The Fed and Central Banks in other countries buy futures contracts on broad market indexes and derivatives on the same. If you Google it, you will find plenty of information.
Certainly, the Fed has expanded its balance sheet with huge amounts of bond buying, forcing money into the stock market and making it attractive for companies to begin stock buy-backs with such low rates.
It goes beyond that though. I recently sold my investment management firm after 27 years of being involved. Every trader I know, who sits at a terminal every day and watches what is happening tells me that an unseen hand is stepping in at every dicey decline to save it from disaster.
Central banks around the world are buying stocks. When you have lots of money, the easiest way to get exposure is with futures on indexes - or other derivative contracts.
As a sample to save you time...
http://www.zerohedge.com/news/2014-08-30/its-settled-central-banks-trade-sp500-futures
http://nypost.com/2014/10/20/plunge-protection-behind-markets-sudden-recovery/
http://www.zerohedge.com/article/us-government-buying-stocks
http://business.time.com/2013/04/26/why-are-central-banks-suddenly-buying-stocks/
http://www.zerohedge.com/news/2015-03-11/how-boj-stepped-143-times-send-japanese-stocks-soaring
We're on the same page with the overseas banks story (from here):
The Bank of Japan, holder of the second-biggest reserves, said April 4 it will more than double investments in equity exchange-traded funds to 3.5 trillion yen ($35.2 billion) by 2014. The Bank of Israel bought stocks for the first time last year while the Swiss National Bank and the Czech National Bank have boosted their holdings to at least 10 percent of reserves.
There's no hard info on the Fed owning common stock, futures, derivatives, or ETF's. That means it'd have to be super secret trades that are invisible to any kind of radar. Kind of like the black helicopters or those mind control cosmic rays we ward off w/ tin-foil hats. So what we're really seriously talking about here is world wide a few billion in funds (say $100B ?) managed by private firms out of a total world wide market cap of maybe $50T.
We need to understand clearly here that $50T is 500 times bigger than $100B and that buying a fifth of a percent does not represent "control".