CAIRO — An obscure militant group based in Egypt’s North Sinai region claimed responsibility over the weekend for a cross-border attack that killed an Israeli soldier last week. The claim called fresh attention to the uphill struggle the newly formed Egyptian government is facing to control the restive Sinai region

The group, which calls itself Supporters of the Holy Places, said in a statement posted on militant Web sites that it mounted the attack in response to a crude American-made video mocking the Prophet Muhammad, the same video cited in anti-American protests around the Muslim world. The group repeated a false claim that a number of Jews were involved in making the video, an apparent lie initially told by a Coptic Christian, an Egyptian expatriate, who did play a role in the production.

The militant group also said it was responsible for a cross-border attack near the Israeli resort of Eilat last year that killed 8 people and wounded more than 30. The Israeli response to that attack inadvertently killed five Egyptian soldiers, bringing the two nations to the brink of a crisis. The militant group said that one of its leaders, Ibrahim Aweida, helped lead the Eilat attack and that Israel killed him last month in the Sinai village of Khreiza; the attack on Friday was partly to avenge his death, the group said.

All three of the attackers were killed by Israeli security forces; their bodies were returned to Egypt over the weekend, state news media reported.

None of the militant group’s statements could be confirmed.

The same group claimed responsibility for bombing a natural gas pipeline to Israel this year; the pipeline was bombed frequently over the last 18 months, until Egypt stopped shipping gas to Israel in a price dispute.

President Mohamed Morsi, a former leader of the Muslim Brotherhood, began a military campaign last month to clear Sinai of what he has called “militants” and “criminals,” after an attack on an Egyptian Army checkpoint killed 16 soldiers. There are often reports of gun battles between Egyptian soldiers and militants, and the army has announced the detention of dozens of suspects and the closing of at least 31 cross-border smuggling tunnels.

But the Sinai region, dominated by Bedouin clans who consider smuggling revenue a birthright, was relatively lawless even under former President Hosni Mubarak’s police state. Residents often refer to Cairo, even Egypt, as a different country or colonial power. After Mr. Mubarak’s government collapsed, the police scattered and lawlessness reached a new peak.

President Morsi told a visiting business delegation that the reassertion of Egyptian “sovereignty” in Sinai was a top priority, participants said, speaking on the condition of anonymity to discuss a private meeting.

But security on the Egyptian side of the border remains a challenge. Last week, Egyptian state news media reported that a government checkpoint on the road to the Rafah border crossing had been attacked by unidentified gunmen for the 36th time since Mr. Mubarak was ousted in February 2011.