Initially, QE contributed to a pretty significant increase in inequality. It raised asset prices, which are owned primarily by the wealthy, while having relatively small if any positive impacts on bank lending, employment, wages or economic growth, so ordinary people haven't had much help. By the third round of QE in 2012-2014, the effects had likely muted quite a bit. There were probably not big impacts on asset prices from QE and the positive effects on employment growth might have strengthened somewhat.

There is no doubt that the Fed's large-scale asset purchases have caused major increases in a number of asset prices in the economy. This is especially true of mortgage backed securities and corporate bonds, and quite possibly of equities as well. For those people and institutions holding those things, the run up in prices has been a wealth bonanza.

The impact of QE on generating more lending by Wall Street to Main Street and in generating more employment and increasing overall investment in the economy is quite modest. QE probably limited the initial collapse of the economy in 2008, and likely had a very small positive impact on economic growth, but its broader impact on jobs and growth in the economy seems not very big.

The Fed cannot reduce inequality on its own; far from it. This requires a concerted effort by the government, broadly speaking, to support a variety of efforts. These include things like raising the federal minimum wage, eliminating unfair restrictions on union organizing, increased fiscal spending on needed infrastructure with a condition that these jobs will be decent paying jobs. Of course, this is just the tip of the iceberg, and far from the question of the role of the Federal Reserve.

Many of those on the right distrust the Fed and want to eliminate its power in the belief that the private economy, including the private banks, will be much more efficient, productive and even democratic if they are left to themselves: in other words, the criticism of the Fed really reflects a desire to cripple the government in the service of increasing the power and authority of the market.

The Fed has a lot of power in the economy because it has a big impact on the supply and cost of credit, that is, interest rates. It also plays a key role in supervising banks and historically has seemed to take it easy on the banks when it shouldn't have, such as in the lead up to the financial crisis.

Increases in interest rates normally worsen inequality, at least partly by reducing employment and wage growth.