Do countries invest in the country's own stock market? For  example can the government of the UK invest in some company with shares in the FTSE 100? Or can Russia take the opportunity now that the shares of oil companies are low to buy them?
One example of monetary policy is the Bank of Japan buying the Nikkei as part of its asset purchasing program.

Typically, a substantial share of citizens' wealth is connected to returns of their national stock market.  Hence, it is generally not a wise move for a fiscal authority to invest public funds (not tied to pension schemes) in broad baskets of stocks.

You might try looking into sovereign wealth funds, these are typically gargantuan profit-maximizing investment vehicles wealthy nations use to acquire risky securities.