WASHINGTON — The United States has carried out at least a dozen operations — including commando raids and airstrikes — in the past three weeks against militants in Afghanistan aligned with the Islamic State, expanding the Obama administration’s military campaign against the terrorist group beyond Iraq and Syria.

The operations followed President Obama’s decision last month to broaden the authority of American commanders to attack the Islamic State’s new branch in Afghanistan. The administration — which has been accused by Republicans of not having a strategy to defeat the group — is revamping plans for how it fights the terrorist organization in regions where it has developed affiliates.

Many of these recent raids and strikes in Afghanistan have been in the Tora Bora region of Nangarhar Province — an inhospitable, mountainous area in the eastern part of the country, near the border with Pakistan. It was in Tora Bora that Osama bin Laden and other senior Qaeda militants took refuge during the American-led invasion in 2001, and eventually evaded capture by slipping into Pakistan.

American commanders in Afghanistan said they believed that between 90 and 100 Islamic State militants had been killed in the recent operations. Intelligence officials estimate that there are roughly 1,000 Islamic State fighters in Nangarhar Province, and perhaps several thousand more elsewhere in the country. But even the generals leading the missions acknowledge that a resilient militant organization can recruit new fighters to replace those killed in American attacks.