The Obama administration insisted Thursday that the U.S. will avoid mission creep in freshly war-torn Iraq, even as its Pentagon chief warned that the Islamic State poses a terrorist threat to Americans “beyond anything that we’ve seen.”

“Get ready,” Defense Secretary Chuck Hagel declared in an ominous warning, even as the White House tried to tamp down speculation that stepped-up airstrikes might lead to a greater ground troop intervention.

Some lawmakers are dubious of the Obama administration’s balancing act, arguing that the stated mission — to protect American personnel in Iraq, provide humanitarian assistance to displaced civilians and aid Iraqi security forces in their fight against Islamist militants — already has evolved from what it was just two weeks ago, when President Obama announced that the U.S. was resuming combat operations in Iraq.

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“The mission already crept a bit,” said Rep. Adam B. Schiff, California Democrat and a member of the House Permanent Select Committee on Intelligence. “The administration would be wise to not get sucked in. That’s going to be very hard.”

In a press briefing at the Pentagon on Thursday, Mr. Hagel dismissed questions about mission creep and pointed to the larger danger that the “barbaric” Islamic State poses.

The group, which is also known by the acronyms ISIL or ISIS, shows a sophistication of terrorism that presents a whole new dynamic and paradigm of threats to the United States, Mr. Hagel said.

“They’re beyond any terrorist group,” he said. “They marry ideology, sophistication of strategic and tactical military prowess. They are tremendously well-funded. Oh, this is beyond anything that we’ve seen, so we must prepare for everything, and the only way you do that is you take a cold, steely hard look at it and get ready.”

Mr. Hagel and Joint Chiefs of Staff Chairman Gen. Martin E. Dempsey addressed reporters’ questions in the wake of more airstrikes and mounting international determination to defeat the group.

Over the span of several days, the U.S. has upped its air campaign in Iraq as it has struggled to wrest the Mosul Dam from the clutches of the Islamic State.

By Tuesday, the number of strikes carried out by U.S. aircraft had tripled from 33 to 90, according to a statement by U.S. Central Command.

Of those 90 airstrikes, 57 were used to support Iraqi and Kurdish fighters battling the Islamic State for control of the dam and surrounding area, CENTCOM said.

Earlier this week, Mr. Obama touted the fact that Iraqi and Kurdish forces, with the help of U.S. air power, reclaimed control of the dam.

But Republicans and Democrats on Capitol Hill have pushed the administration to do more, including launching airstrikes against militant forces in Syria.

Syrian President Bashar Assad’s forces Thursday killed dozens of Islamic State members in heavy fighting in the northern province of Raqqa, where the Islamic State holds considerable territory. Two years ago, it took captive American photojournalist James Foley while he was covering the Syrian civil war.

Within 24 hours of Mr. Obama’s declaration that Iraq’s Mosul Dam had been retaken, the Islamic State countered by releasing a video showing Mr. Foley being beheaded and threatening to kill another reporter held captive.

Broadcasting the death of Mr. Foley was just one of the many horrific acts perpetrated by the Islamic State. The group also has slaughtered Christians and other religious minorities across Iraq and is responsible for widespread acts of violence and thuggery.

The White House revealed Wednesday that a mission to rescue American captives held by the Islamic State this summer had failed. Mr. Hagel on Thursday cast the mission as a “flawless operation,” even though it was not successful.

“It’s the responsibility of our government and our leaders to do all we can to take action when we believe there might be a good possibility, a good chance to make a rescue effort successful,” he said.

“This operation, by the way, was a flawless operation. But the hostages were not there. So we’ll do everything that we need to do, that the American people would expect from their leaders, to continue to do everything we can to get our hostages back,” he said.

Some military analysts say the Foley video could provide the White House and Pentagon with justification for further action against the Islamic State should they choose that route. Both Republicans and Democrats on Capitol Hill continue to call on the president to launch a broader campaign against the terrorist organization.

“I think ISIL made a huge strategic blunder with the video of the execution of the American. That really does provide a new and graphic basis for the president to take action against ISIL. In other words, it’s inarguable they present a threat to America,” said Charlie Dunlap, a law professor at Duke University who retired in 2010 as a major general in the U.S. Air Force Judge Advocate Corps.

“I think the video provides political leaders not just in the U.S. but elsewhere with the ready rationale for their publics. It’s very easy to convey the threat ISIL presents when you have a video like that,” he said.

Speaking at the conservative Heritage Foundation on Thursday, Texas Gov. Rick Perry said the U.S. should be open to sending heavy weapons to Kurdish forces and deploying more American ground troops.

“We need to have all of our options open,” the governor said. “Just about the last thing we want to think about is more conflict in Iraq and what that might involve. But we’d better get on top of this crisis by any means necessary.”

The U.S. has 750 military personnel in Iraq in addition to the roughly 100 who work out of the U.S. Embassy in Baghdad.

Pentagon officials said Thursday that if the U.S. military’s air campaign in Iraq continues to move forward and its humanitarian assistance endeavor expands into an international effort, the U.S. will look to its partners for direction and is not interested in spearheading an international coalition.

“Every time we have a coalition, the U.S. doesn’t have to lead,” a senior Pentagon official said.

A second senior Pentagon official concurred, saying the role of the United States in Iraq was projected to remain limited.

“The U.S. isn’t leading a campaign inside Iraq,” the official said. “This isn’t 2003,” when President George W. Bush ordered the invasion of Iraq to overthrow the Saddam Hussein dictatorship.

Faced with a much different enemy a decade later, Mr. Obama has stressed that the international community, along with the fledgling Iraqi government, must step up to confront the Islamic State.

“From governments and peoples across the Middle East, there has to be a common effort to extract this cancer so that it does not spread. There has to be a clear rejection of these kind of nihilistic ideologies,” he said. “We share a common security and a common set of values that are rooted in the opposite of what we saw yesterday [on the Foley video]. And we will continue to confront this hateful terrorism and replace it with a sense of hope and civility.”

Gen. Dempsey said it will take an international coalition to defeat the “apocalyptic, end-of-days strategic vision” of the terrorist group, whose members also have taken over a portion of Syria. It is impossible to defeat the Islamic State “without addressing that part of the organization that resides in Syria,” Gen. Dempsey said.

“That will have to be addressed on both sides of what is essentially, at this point, a nonexistent border,” he said. “And that will come when we have a coalition in the region that takes on the task of defeating ISIS over time. It requires a variety of instruments. Only one small part of it is airstrikes.”

Five NATO members — Britain, Canada, France, Germany and Italy — have offered to deliver weapons to the Kurdish fighters at the forefront of the bloody battle. Australia also has offered to help, but only by delivering humanitarian aid.

• Stephen Dinan contributed to this article, which is based in part on wire service reports.

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