Just two days after Speaker John A. Boehner stunned Washington by announcing he will leave Congress next month, two top members of his House Republican conference traded barbs in a remarkably public display of internal dissent on a Sunday network news show.

Deputy Whip Tom Cole, R-Okla., and House Freedom Caucus founding member Mick Mulvaney, R-S.C., did agree on two things. They both see Majority Leader Kevin McCarthy, R-Calif., as the leading candidate to take over as speaker. And they believe a government shutdown will be averted by a stopgap spending bill passed within the next few days.

“I don’t think there’s much doubt” of McCarthy winning a coming speaker election, Cole said on “Fox News Sunday.” Mulvaney agreed, saying McCarthy “has the inside track.”

But the agreement ended there, with Cole and Mulvaney spending much of their joint appearance arguing even as some suggest Boehner’s decision to remove himself from the party’s years-long internal fight would bring peace to the GOP caucus.

Mulvaney made clear House conservatives want the next speaker to be more aggressive in countering the Obama administration, which he described as “overreaching” on policy matters and going around Congress to enact its agenda.

“We stopped being a coequal branch of government,” Mulvaney said when asked by host Chris Wallace “what went wrong,” in his view, under Boehner, R-Ohio.

“Congress used to use the power of the purse  . . .  against an overreaching administration. We stopped,” he said. “It had to change.”

The duo bickered about a list of issues, including Congress’ track record at overseeing the executive branch and Planned Parenthood funding.

Mulvaney criticized Boehner and other House GOP leaders for “overpromising” on policy and legislation but “underperforming.”

On ABC’s “This Week” program, Republican presidential candidate Ben Carson struck a similar tone. He called Boehner’s resignation a good thing because it is time for “new leadership” in Washington. Carson said Republican voters are frustrated with the members they have elected in recent years “and they want to see some results.”

In a statement released earlier Sunday morning, the conservative Freedom Caucus, which organized in part to oppose Boehner, said it is not ready to endorse a new speaker. The group also said it intends to meet with each leadership candidate.

The House has yet to set a date for the election of new leaders. But Cole said he wants them to happen soon, saying he sees little use in “dragging this out.”

Appearing a bit later on “Face the Nation” on CBS, Boehner joined Cole and Mulvaney in predicting Congress later this week will pass a short-term funding measure to avert a government shutdown.

With four calendar days remaining before the federal lights would go dark, Boehner told CBS News that he intends to pass the  continuing resolution with help from Democrats before a Wednesday night deadline. Boehner, who announced his retirement Friday, did not indicate how long the CR would span.

A Senate version ( HR 719 ) would keep the government open until Dec. 11.

Boehner, who teared up several times during the interview, said he believes House Democrats want to avoid a shutdown “as much as I do.”

Many lawmakers believe Boehner made his stunning decision to leave Congress largely because of ongoing clashes with the House Freedom Caucus. On Sunday, he criticized those conservative members for demanding legislative actions “that are never going to happen.”

Boehner ticked off a list of what he called his biggest accomplishments as speaker, including a deficit-reduction deal, tax cuts and entitlement reforms. He criticized the rebellious faction of his caucus for opposing each one.

“All voted against by my conservatives members,” Boehner said. “Really?!”

Boehner offered some advice to his yet-unknown replacement: “Have the courage to do what you can,” adding the next speaker should understand that “Hail Mary passes” do not work in the U.S. legislative system.