In a new book, the anti-Trump Republican Liz Cheney accuses Mike Johnson of dishonesty over both the authorship of a supreme court brief in support of Donald Trump’s attempt to overthrow the 2020 election and the document’s contents, saying the congressman, who is now US House speaker, duped his party with a “bait and switch”.

“As I read the amicus brief – which was poorly written – it became clear Mike was being less than honest,” Cheney writes. “He was playing bait and switch, assuring members that the brief made no claims about specific allegations of [electoral] fraud when, in fact, it was full of such claims.”

Cheney also says Johnson was neither the author of the brief nor a “constitutional law expert”, as he was “telling colleagues he was”. Pro-Trump lawyers actually wrote the document, Cheney writes.

In a statement to the Guardian, a spokesperson for Johnson said Cheney was “not presenting an accurate portrayal”.

As Trump’s attempts to overturn his defeat by Joe Biden progressed towards the deadly January 6 attack on Congress, Cheney was a House Republican leader. Turning against Trump, she sat on the House January 6 committee and was ostracised by her party, losing her Wyoming seat last year.

Her book, Oath and Honor: A Memoir and a Warning, will be published next week. The Guardian obtained a copy.

Johnson became speaker last month, after McCarthy was ejected by the Trumpist far right, the first House speaker ever removed by his own party.

On Tuesday, CNN ran excerpts from Cheney’s book, quoting her view that Johnson “appeared especially susceptible to flattery from Trump and aspired to being anywhere in Trump’s orbit”.

CNN also reported that Cheney writes: “When I confronted him with the flaws in his legal arguments, Johnson would often concede, or say something to the effect of, ‘We just need to do this one last thing for Trump.’”

But Cheney’s portrait of Johnson’s manoeuvres is more comprehensive and arguably considerably more damning.

The case in which the amicus brief was filed saw Republican states led by Texas attempt to persuade the supreme court to side with Trump over his electoral fraud lies.

It did not. As Cheney points out, even the two most rightwing justices, Samuel Alito and Clarence Thomas, who wanted to hear the case, said they would not have sided with the complainants.

Cheney describes how Johnson, then Republican study committee chair, emailed GOP members on 9 December 2020 to say Trump had “specifically” asked him to request all Republicans in Congress “join on to our brief”.

Johnson, Cheney says, insisted he was not trying to pressure people and simply wanted to show support for Trump, by “affirm[ing] for the court (and our constituents back home) our serious concerns with the integrity of our electoral system” and seeking “careful, timely review”.

“Mike was seriously misleading our members,” Cheney writes. “The brief did assert as facts known to the amici many allegations of fraud and serious wrongdoing by officials in multiple states.”

Johnson, she says, then told Republicans that 105 House members had expressed interest. “Not one of them had seen the brief,” Cheney writes. She also says he added “a new inaccurate claim”, that state officials had been “clearly shown” to have violated the constitution.

“But virtually all those claims had already been heard by the courts and decided against Trump.”

Calling the brief “poorly written”, Cheney says she doubted Johnson’s honesty and asked him who wrote it, as “to assert facts in a federal court without personal knowledge” would “present ethical questions for anyone who is a member of the bar”.

The general counsel to McCarthy, then Republican minority leader, told Cheney that McCarthy would not sign the brief, while McCarthy’s chief of staff also called it “a bait and switch”. McCarthy told her he would not sign on. When the brief was filed, McCarthy had not signed it. But “less than 24 hours later, a revised version … bore the names of 20 additional members. Among them was Kevin McCarthy.

“Mike Johnson blamed a ‘clerical error’ … [which] was also the rationale given to the supreme court for the revised filing. In fact, McCarthy had first chosen not to be on the brief, then changed his mind, likely because of pressure from Trump.”

It took the court a few hours to reject the Texas suit. But the saga was not over. Trump continued to seek to overturn his defeat, culminating in the deadly attack on Congress on 6 January 2021 by supporters whom he told to “fight like hell”.

Cheney takes other shots at Johnson. But in picking apart his role in the amicus brief, she strikes close to claims made for his legal abilities as he grasped the speaker’s gavel last month. Johnson “was telling our colleagues he was a constitutional law expert, while advocating positions that were constitutionally infirm”, Cheney writes.

Citing conversations with other Republicans about Johnson’s “lawsuit gimmick” (as she says James Comer of Kentucky, now House oversight chair, called it), Cheney says she “ultimately learned” that Johnson did not write the brief.

“A team of lawyers who were also apparently advising Trump had in fact drafted [it],” she writes. “Mike Johnson had left the impression that he was responsible for the brief, but he was just carrying Trump’s water.”

A spokesperson for Johnson said: “Unfortunately, Ms Cheney is not presenting an accurate portrayal of those events. And while he does not plan to purchase a copy of Oath and Honor, Speaker Johnson wishes former representative Cheney and her family the best in whatever her future endeavours may be.”

Earlier, responding to CNN, a Trump spokesperson said Cheney’s book belonged “in the fiction section of the bookstore”.

Cheney also considers the run-up to January 6 and the historic day itself. Before it, she writes, she and Johnson discussed mounting danger of serious unrest. He agreed, she says, but cited support for Trump among Republican voters as a reason not to abandon the president. Such support from Johnson and other senior Republicans, Cheney writes, allowed Trump to create a full-blown crisis.

Two and a half years on, notwithstanding 91 criminal charges, 17 for election subversion, Trump is the clear frontrunner for the Republican presidential nomination. He polls close to or ahead of Biden.

In certain circumstances, close elections can be thrown to the House – which Mike Johnson now controls.