While Trump and Barr have focused on Antifa, the FBI and other agencies are tracking groups from both the extremist right and left involved in the riots and attacks on police.
Federal law enforcement officials tell CNN they are aware of organized groups who are seeking to carry out the property destruction and violence, using the cover of the legitimate protests in Minneapolis and elsewhere. Those domestic extremist groups include anarchists, anti-government groups often associated with far-right extremists and white supremacy causes, and far-left extremists who identify with anti-fascist ideology.
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The FBI and other federal agencies are working with local authorities to track social media posts and other communications to identify those who have crossed state lines to carry out violence. Federal officials say they believe there is an amalgam of groups showing up in protests from the extremist left and right that normally oppose each other but see common cause in attacking the police and the government.
The Trump administration's message has focused only on the far-left extremists. Barr's statement referred to violence from "antifa and other similar groups." On CNN's "State of the Union" on Sunday, national security adviser Robert O'Brien said, "These Antifa groups are organized and use Molotov cocktails and fireworks and gas to burn down our cities."
Asked by CNN's Jake Tapper about far-right groups participating in the violence, O'Brien said: "I haven't seen the reports on far-right groups. This is being driven by Antifa."
Trump has also blamed Democratic officials in Minnesota and other states where violence has occurred. "Get tough Democrat Mayors and Governors. These people are ANARCHISTS. Call in our National Guard NOW," Trump tweeted Sunday.
Acting Senate Intelligence Chairman Marco Rubio, a Florida Republican, said Saturday that the outsiders included "a rogues gallery of terrorists from Antifa to 'Boogaloo' groups encouraging and committing violence." Boogaloo is a group often associated with far-right extremist ideology that wants to initiate a civil war. Rubio tweeted that the groups were not "ideologically compatible but share a hatred of govt & police & are taking advantage of the protests."
"Many of these professional agitators don't fit a simple left vs right identity," Rubio said. "They are part of a growing anti-government extremist movement. They hate law enforcement & want to tear the whole system down even if it requires a new civil war."

What local officials are saying
Minnesota officials said this weekend white supremacists and others were mixing in with legitimate protestors. Authorities there are looking at connections between those arrested and white supremacist organizers who have posted online about coming to Minnesota. Officials have also tracked messages about seeking to loot and whether there was a connection to organized crime.

At a news conference on Sunday morning, Minneapolis Public Safety Commissioner John Harrington said that one-fifth of the weekend arrests to date were from outside the state. Among those arrested Saturday were residents of Arkansas, Kansas City, Iowa, Illinois and Michigan, Harrington said, though he did not have arrest data from overnight.
Walz also said the state was looking at who was behind a "very sophisticated" denial of service attack on all state computers that was executed on Saturday. "That's not somebody sitting in their basement, that's pretty sophisticated," he said.

Officials across the country are also assessing the make-up of the agitators mixed in with legitimate protestors.
Denver Mayor Michael Hancock told CNN's Wolf Blitzer on Sunday he "wouldn't be surprised if Antifa" was behind some of the actions in his city. "We don't have the specific information they're directly engaged, but we have intel, we have watched and intercepted, frankly, groups coming into Denver. We have confiscated weapons, including assault weapons, that were heading to the demonstrations."
Salt Lake City Mayor Erin Mendenhall said the city too is trying to determine the role of those behind the violence, telling CNN, "We have yet to unpack of the arrests we made who the instigators are and their locations. We're anxious to find out."

In Chicago, Mayor Lori Lightfoot said looters organized across the city and possibly came in from outside: "There clearly was coordination, they were clearly listening to our radio traffic," she said. "The number of U-Haul trucks that magically showed up in front of stores, car caravans that dropped people off and broke windows, and then were hustling the goods out into the backs of the cars. Absolutely, it was organized -- there's no question whatsoever about that."

Philadelphia Mayor Jim Kenney on Sunday denounced what he called "anarchists" for the looting and vandalism that occurred Saturday night in his city and claimed rhetoric from the White House was "perpetuating" the violence.
In New York, investigators are also searching social media to identify leaders who are trying to get protestors to gather to incite violence, a law enforcement official told CNN. Investigators there have seen antagonizers more affiliated with Antifa and far-left causes, and not necessarily white supremacist groups. "They do not represent the peaceful New Yorkers that want to protest," the official said.

Who should do that? There is no such thing as "The Antifa" or a central leadership who decides on all. It is just a whole bunch of individuals and local groups with the same ideas waving the same flag, it's a movement not an organisation and nobody is in the position to speak on behalf of the movement.