By Alexis Benveniste, CNN Business

(CNN) -- Congresswoman Marjorie Taylor Greene, a Trump-backed QAnon supporter, is a fountain of misinformation, conspiracy theories and hateful rhetoric, and most mainstream media outlets are covering the controversy. But Fox News is still wrapped up in attacking Congresswoman Liz Cheney.

Cheney, a Wyoming congresswoman and the No. 3 House Republican, has been the target of both Republican and right-wing media backlash since she voted to impeach Former President Donald Trump following the January 6 Capitol siege.

"It's all about punishing Cheney; it's about punishing dissenters who dare — who dare — to impeach the former president," CNN's Chief Media Correspondent Brian Stelter said on "Reliable Sources" Sunday. "And the idea of a Senate trial — it's crazy! To even think about trying Trump is crazy in right-wing media."

Although right-wing outlets refuse to take their eyes off of the Cheney ball, Greene continues to spread misinformation. On January 17, Twitter took note of Greene's rhetoric, temporarily suspending her from the social media platform after she tweeted a conspiracy-laden thread about the Georgia Senate elections.

"You'll notice that in pro-Trump outlets, that Greene is barely getting attention," said Stelter. "Their eyes instead are set on Congresswoman Liz Cheney."

On Fox News Friday night, anchor Shannon Bream spoke about Cheney over a chyron that read, "Canceling Liz Cheney."

"Greene is getting attention from traditional outlets — she's being scrutinized — but Cheney is being torn down from pro-Trump media," Stelter said, adding, It's important to recognize that Congresswoman Greene is just a small piece of a larger movement, of a larger phenomenon."

The larger phenomenon is the spread of the baseless QAnon theory, a set of views that has been backed by many of the people who stormed the Capitol earlier this month and defended by the likes of Tucker Carlson.

Last week, CNN reported that Greene repeatedly showed support for executing executing prominent Democratic politicians in 2018 and 2019 before she was elected to Congress. And on Saturday, she tweeted: "I won't back down. I'll never apologize. And I'll always keep fighting for the people."

Speaker Nancy Pelosi last week slammed House Republican leaders for not doing more to rebuke Greene over her shocking rhetoric and past statements.

"They are coming after me because, like President Trump, I will always defend conservative values," Greene said last week. "They want to take me out because I represent the people. And they absolutely hate it."