FRANKFURT — Martin Winterkorn was uncharacteristically subdued last week when, on the eve of the Frankfurt auto show, he stepped into a beam of blue light to show off the latest products from the Volkswagen empire.

The chief executive certainly knew what was coming days later: allegations that Volkswagen had cheated to pass emissions tests in the United States, touching off a scandal that has called his leadership into question and seriously damaged the automaker’s reputation.

Since the news broke on Friday, Volkswagen has scrambled to control the damage.

Mr. Winterkorn issued an apology on Sunday, saying the company had “broken the trust of our customers and the public.” Volkswagen said it would stop selling the remainder of its 2015 model Volkswagen and Audi diesels and not offer its 2016 diesel cars, which were just arriving in showrooms in the United States.

These measures did not stop Volkswagen shares from plummeting more than 20 percent when trading opened in Europe on Monday. Analysts say that Mr. Winterkorn will have to answer tough questions — including when he and other top executives at Volkswagen, which is known for being a tightly controlled, autocratic organization, first learned of the deception.