WASHINGTON—In claiming responsibility for Sunday's deadly bombings in Uganda, the Somali group al Shabaab appeared to raise its sights from running large swaths of one of Africa's most lawless countries to embracing al Qaeda's call for global jihad.

The triple bombing in Kampala, which left 76 dead, would mark the first foray beyond war-torn Somalia by a group whose primary focus for years was to impose a radical Islamist agenda inside Somalia. The attacks show what Africa and terror experts say is the growing influence in the group of foreign fighters with ties to Osama bin Laden's global terror network.

Al Shabaab was formally created in about 2003 as a successor movement to violent Islamist groups that had been crushed in the late 1990s. It is now the target of an offensive by Somalia's elected government, which controls a sliver of Somalia's capital.

In recent years, al Shabaab has claimed to have moved closer to al Qaeda. In late 2008, al Shabaab leaders pledged fealty to Mr. bin Laden. Early the next year, al Qaeda welcomed al Shabaab to its global fight. Earlier this year, al Shabaab announced it was formally linking its local fight to al Qaeda's global jihad.

"The links between al Qaeda and al Shabaab are stronger," said David Shinn, the former U.S. ambassador to Ethiopia and an expert on al Shabaab. He and a U.S. counterterrorism expert said the two groups work together to train operatives.

Al Shabaab's force numbers a few thousand fighters, by some estimates, to more than 10,000. A contingent of foreign fighters is said to number a few hundred and include other Africans, Arabs and even Westerners.

In late 2008, a score of Somali-Americans left Minnesota to join al Shabaab. Last month, two men were arrested at JFK International Airport allegedly on their way to join the Somali group. One of al Shabaab's top commanders, known as Abu Mansoor al-Amriki, is from Alabama.

Top commanders have spent time in Afghanistan and Pakistan, says Rashid Abdi, Horn of Africa analyst in Nairobi for the International Crisis Group. "What al Shabaab has become is a multinational terror network, like al Qaeda if not al Qaeda," he says.

Analysts say two factions have emerged within the group. One wants to stay focused on bringing down the government, a goal for which they'll need to retain some support among Somalia's population. Those aligned with al Qaeda don't share those priorities, says Anneli Botha, a senior researcher on terrorism at the Institute for Security Studies, a think tank based in Pretoria.

But to some extent there is an alignment of goals between Somali nationalists with an eye on regional expansion and jihadists eager to strike out at perceived regional enemies that include Christians, other Muslims, moderate politicians and supporters of Somalia's interim government.

In recent months, al Shabaab had threatened countries, including Uganda, that contribute peacekeeping troops in support of the Somali transitional government. The U.S. is the major backer of the African peacekeeping troops.

"This is really an unpleasant confluence of goals between the nationalist and international wing of al Shabaab," said Roger Middleton, a Somalia analyst at London-based think tank Chatham House.

"Regionalization of the conflict is a rational strategy for al Shabaab. They are stronger when there are foreign boots on the ground, and weaker otherwise," said William Braniff, an instructor at the Combatting Terrorism Center at West Point.

The links between al Shabaab and al Qaeda—especially al Qaeda's affiliate in Yemen—could pose the biggest threat to the U.S. Al Qaeda in the Arabian Peninsula, as the Yemeni affiliate is known, was behind several recent high-profile attacks on the U.S., including the failed Christmas Day bombing of a Detroit-bound airliner.

Siobhan Gorman, Peter Wonacott and Nicholas Bariyo contributed to this article.