Linda Sarsour awoke on Jan. 23, 2017, logged onto the internet, and felt sick. The weekend before, she had stood in Washington at the head of the Women’s March, a mobilization against President Donald J. Trump that surpassed all expectations. Crowds had begun forming before dawn, and by the time she climbed up onto the stage, they extended farther than the eye could see.

But then something shifted, seemingly overnight. What she saw on Twitter that Monday was a torrent of focused grievance that targeted her. In 15 years as an activist, largely advocating for the rights of Muslims, she had faced pushback, but this was of a different magnitude. A question began to form in her mind: Do they really hate me that much?

That morning, there were things going on that Ms. Sarsour could not imagine. More than 4,000 miles away, organizations linked to the Russian government had assigned teams to the Women’s March. At desks in bland offices in St. Petersburg, using models derived from advertising and public relations, copywriters were testing out social media messages critical of the Women’s March movement, adopting the personas of fictional Americans.