Problem

Over the years, economists, universities, and investors criticized the Fed for keeping the CPI levels too low to reflect actual consumer market prices. Despite the inflation rate at the 30-year high (6.2%), some goods and services prices increased significantly more.
The seasonally adjusted CPI for Urban Consumer for chosen goods and services based on BLS data (November, 2021). Source: https://www.visualcapitalist.com/u-s-inflation-which-categories-have-been-hit-the-hardest/

The Fed has the means to influence the data

There are many ways in which the Feb and BLS can influence the CPI. Most common are substituting the underlying data selection and their relative importance in the basket of goods.
Under Congressional pressure, BLS has also frequently changed the CPI calculation methods. And the Fed altered the reporting frequency and discontinued certain metrics.
For example, in 2020, as the money supply increased appr. 40% YOY, the Fed changed its M1 and M2 indexes, discontinued the old metrics, and decreased reporting frequency to once a month.
Relationship between M2 money supply per capita growth and the CPI inflation assembled by Lyn Alden based on the Federal Reserve data (October 2021). Source: https://www.lynalden.com/november-2020-newsletter/
The changes reflected the Fed’s Chair, Jerome Powell’s, belief that money supply doesn’t cause inflation, which many economic schools, researchers, and the historical cases of hyperinflation do not support.

The Fed has the incentive to misinform the public.

Inflation is a highly political subject reported by a subjective and centralized entity in a semi-transparent manner. The BLS is an information-gathering unit for the Department of Labor and the Federal Reserve, subject to influences from Congress, the Fed, and the US Treasury. Its quality of work also directly depends on the federal government budget.
Some incentives to keep the reported inflation rate low: Maintaining trust in the US dollar as the world’s reserve currency Higher negotiating power with other countries Monetary and fiscal policies (keeping interest rates low not to thwart economic growth and default on the massive US debt) Preventing stock markets volatility The perceived success of the residing government and the president Lower social programs and retirement payouts Preventing panic and the inflation loop
“There is no way we can survive the cure for inflation [higher interest rates], the cure for inflation would kill the economy, which is why we’re not gonna get it. Which is why inflation is here to stay and is gonna keep getting worse” Peter D. Schiff

The BLS uses outdated data collection methods and data sources

The CPI is usually calculated using data from door-to-door census or brick-and-mortar surveys at the point of purchase (PoP) and, most recently, phone surveys due to the pandemic. BLS workers painstakingly collect the data over the years and slowly input it into the legacy system. Many data sets are older than 2-years.
Recently, the researchers from Penn State University raised doubts about rent data collected by surveying house owners and renters. The rent prices that have the greatest relative importance in the CPI basket rely on outdated, self-reported, and skewed information that does not reflect the rapid increases in the market.

No one has access to the complete data

Researchers and economists battle over inflation theories but never reach a consensus because they lack the raw data and the situation overview, sole purview of the Fed or central banks.
Many researchers pointed out the problem and led MIT and Harvard economists Alberto Cavallo and Roberto Rigobon to sponsor The 1 Billion Prices Project, collecting daily product prices independently of the government between 2008 and 2016.