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

QE and other aspects of Fed policy increased inequality pretty significantly. This is reinforced if you take into account all the other non-standard measures the Fed used to bail out the banks early on in the [2008] crisis.

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.

The right wants to destroy the power of the Fed to increase the power of finance; and the progressives want to reorient the Fed so that it will stop protecting the interests of finance and protect the interests of the broader population instead.

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.

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.

People in America get really angry at the Federal Reserve and at the "money system" in general during economic crises. The Fed draws hostility because of its power, its insulation from democratic accountability, its lack of transparency, and because of its historical and structural connections to finance.