The White House doubled down Thursday on President-elect Donald Trump’s purported knowledge of Russia’s hacking during the election, saying Mr. Trump should “cooperate” with a government investigation into whether Moscow’s interference helped him to win the presidency.

White House press secretary Josh Earnest said that rather than attacking the credibility of U.S. intelligence agencies, Mr. Trump should “actually be supportive of a thorough, transparent, vigorous, non-political investigation into exactly what happened, and to cooperate with it.”

President Obama’s spokesman repeated the administration’s accusation that Mr. Trump “obviously knew that Russia was engaged in malicious cyberactivity that was helping him and hurting Secretary [Hillary] Clinton’s campaign.”

Trump adviser Kellyanne Conway said Thursday that Mr. Earnest’s comments are “irresponsible.”

“He essentially stated that the president-elect had knowledge of this, maybe even fanned the flames. It’s incredibly irresponsible and I wonder if his boss, President Obama agrees,” she said on “Fox & Friends.”

For his part, Mr. Trump again raised doubts about the intelligence community’s assessment that Russia meddled in the election.

“If Russia, or some other entity, was hacking, why did the White House wait so long to act? Why did they only complain after Hillary lost?” Mr. Trump tweeted Thursday.

The president-elect also slammed Mr. Earnest at a rally Thursday night in Hershey, Pennsylvania.

“This foolish guy, Josh Earnest, I don’t know,” Mr. Trump told the crowd. “You know, having the right press secretary is so important because he’s so bad the way he delivers a message. He can deliver a positive message and it sounds bad. He could say ‘ladies and gentlemen, today we have totally defeated ISIS,’ and it wouldn’t sound good, OK?”

Mr. Trump added, “Maybe he is getting his orders from somebody else?”

Conclusions by private analysts about alleged Russian hacking were known last May. White House officials say the administration held back with accusations until October because it took time for 17 government intelligence agencies to reach a consensus, and because Mr. Obama didn’t want to appear to be influencing the agencies’ work or meddling in the election.

Mr. Earnest also ridiculed the Trump team’s explanation that Mr. Trump was being sarcastic in July when he suggested publicly that Russia should locate the emails missing from Mrs. Clinton’s private server.

“You all have it on tape,” Mr. Earnest told reporters. “The Republican nominee for president was encouraging Russia to hack his opponent because he believed that would help his campaign. I don’t think anybody at the White House thinks it’s funny that an adversary of the United States engaged in malicious cyberactivity to destabilize our democracy. That’s not a joke. Nobody at the White House thought it was a joke. Nobody in the intelligence community thought it was a joke.”

The president’s spokesman edged closer to the language of conspiracy when talking about Mr. Trump’s actions.

“It was obvious to everyone who was paying attention, including the gentleman whose thumbs authored that tweet, that the impact of that malicious activity benefited the Trump campaign and hurt the Clinton campaign,” Mr. Earnest said. “That is, after all, why the president-elect called on Russia to hack Secretary Clinton’s email. That is presumably why the coverage of the hack-and-leak operation that Russia carried out was focused on emails from the Democratic Party and Clinton campaign staffers and not the Republican Party and Trump campaign staffers.”

During a news conference on July 27, in the midst of the Democratic National Convention, Mr. Trump said, “I will tell you this, Russia: If you’re listening, I hope you’re able to find the 30,000 emails that are missing. I think you will probably be rewarded mightily by our press.”

His barb came amid persistent questions about Mrs. Clinton’s careless handling of classified emails, and about what Mrs. Clinton claimed was the harmless deletion of thousands of the emails from the private server that she used while working as secretary of state.

At the time, Trump ally Newt Gingrich said that Mr. Trump’s suggestion about Russia was a joke and that the media’s focus should be on Mrs. Clinton’s use of a private email server.

“The media seems more upset by Trump’s joke about Russian hacking than by the fact that Hillary’s personal server was vulnerable to Russia,” Mr. Gingrich said on Twitter.

In addition, Russia already had done whatever hacking it did, and Mrs. Clinton’s server was wiped, offline and in FBI custody.

Mr. Earnest speculated that Mr. Trump learned of the Russian cyberattacks by “relying on news reports, [or] maybe somebody on Capitol Hill who had been briefed about this matter had informed him or his team about it.

It’s also possible he consulted with one of his closest aides, Roger Stone, who back in July — July 27, to be precise — tweeted, quote, ‘Of course the Russians hacked at Hillary Clinton’s e-mail,’” Mr. Earnest said.

Mr. Stone is a longtime Trump friend who left the campaign last year and does not hold a position on the transition team. In October, he claimed to have “back-channel communications” with WikiLeaks about thousands of stolen emails it was preparing to release from the account of Clinton campaign chairman John Podesta.

Soon after that, WikiLeaks began publicizing the emails from Mr. Podesta’s account, providing a steady stream of negative news coverage about the Democratic nominee. Mr. Stone said in a tweet this week that U.S. intelligence agencies have “no proof” of Russian hacking in the election “because it’s bullsh—.”

Craig Murray, a former British envoy with connections to WikiLeaks founder Julian Assange, said this week that he received the damaging emails from a “disgusted” Democratic insider in Washington during a clandestine package drop in September. He said the Democratic whistleblowers were motivated by anger about the corruption of the Clinton Foundation and about efforts by Democratic Party officials to sway the presidential primary away from Sen. Bernard Sanders of Vermont.

Ms. Conway accused Mr. Earnest of “basically trying to relitigate a political campaign” when both teams “are trying to work very closely to have a peaceful transition of power in a great democracy with just about a month-plus to go.”

Despite three consecutive days of lashing out at Mr. Trump over the Russia allegations, Mr. Earnest said the White House has fulfilled its responsibility for a “smooth and effective” transition to Mr. Trump’s inauguration in a month.

“There are others on the outside who are raising these questions, and apparently that is striking a nerve with the president-elect’s team,” he said. “One way to deal with that is to start answering these questions and not just relying on a defense suggesting that the rhetoric of the Republican nominee was a joke, when nobody thought it was funny.”

Asked about Mr. Trump’s tweet on Thursday, transition communications director Jason Miller said he would “let the president-elect’s tweets speak for themselves.”

“But I’d say the continued efforts to try to delegitimize the election, at a certain point you’ve got to realize that the election from last month is going to stand, whether it’s the recount or continued questions along this line,” he told reporters. “And we’re moving ahead and putting together a successful administration that’s ready to go to work serving the American people.”

The administration has yet to provide evidence that Russia was behind the hacking of Democratic officials’ emails and other intrusions into the U.S. election system. The U.S. intelligence agencies put out a joint statement on Oct. 7 that Russia was trying to undermine U.S. democracy and that “only Russia’s senior most officials could have authorized these activities.”

Democrats have said Russian President Vladimir Putin wanted to damage the candidacy of Mrs. Clinton, who infuriated him in 2011 during her tenure as secretary of state by asserting that a Russian election was rigged.

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