They have been working on their English-language music with Dave Sitek of TV on the Radio and the producer Ricky Reed, Ms. Tolokonnikova said.

The last video they released, in late October, was called “Make America Great Again.” It showed fictional Trump agents in red armbands raping and torturing in a campaign against Muslims, Mexicans, women who have abortions, gays and lesbians.

It was certain to offend. But it wasn’t illegal, at least not here — at least not yet.

And it was a modest Russia-in-America answer to the more voluminous pro-Trump propaganda Mr. Putin exported to the United States. Some arrived through his sophisticated state-financed news networks (one, Sputnik, featured #CrookedHillary hashtags on its Twitter feed). And if assessments by the United States intelligence community are correct, some came through state-supported internet skulduggery.

Ms. Tolokonnikova said she became more involved here because the stakes were bigger than one country.

“What happens in one country makes huge influence on what’s going on in other countries,” she said. “So, I didn’t want Donald Trump to be elected because it would obviously encourage authoritarian politicians around the world to be more authoritarian, and it did.” (To wit, President Rodrigo Duterte of the Philippines claiming without substantiation that Mr. Trump had endorsed his murderous drug crackdown.)

Yet as the web enables Mr. Putin to spread propaganda that encourages nationalist movements to campaign for walls and isolation — most recently, it is claimed, in Italy, where a referendum was held on Sunday — it also breaks down the cultural barriers between countries.

There are places in Russia where the internet provides a rare route to real news, given that Mr. Putin has effectively pressured so much of Russia’s independent journalism out of existence on television, on radio and in print.