Prosecutors have said that extra testing time — permitted with guidance from a psychologist — made it easier for Mr. Singer to arrange the cheating by bribing certain test administrators. Yet, in Ms. Huffman’s case, Eric S. Rosen, the lead prosecutor, told the judge that the government was not asserting that extra time had been inappropriately awarded to her daughter.

“We have been working with a neuropsychologist since my daughter was 8 years old and she began receiving extra time on tests since she was 11,” Ms. Huffman said, after asking U.S. District Court Judge Indira Talwani whether she could clarify a few points.

Judge Talwani asked what else she wanted to clarify.

“I think — ” Ms. Huffman began, before her voice gave out and she had to stop for a moment. “I’m sorry — I think you covered it.” She said that she wasn’t knowledgeable about all the details Mr. Rosen had mentioned during the hearing, involving the amounts of money Mr. Singer had paid the test administrator and proctor involved in the cheating.

“But everything else that Mr. Rosen said, I did,” she said.

Ms. Huffman and another parent charged in the case, Devin Sloane, pleaded guilty to the same felony count during a joint court hearing on Monday afternoon. Mr. Sloane was charged with agreeing to pay Mr. Singer $250,000 so that his son would be recruited to the men’s water polo team at the University of Southern California, ensuring his admission, even though his son didn’t actually play the sport. Falsifying sports qualifications was another element of the admissions scheme Mr. Singer arranged, along with cheating on tests, prosecutors say.

An element of the case that remained unanswered after the hearing was Mr. Macy’s role in what happened. Mr. Macy has not been charged in the case. But references in the prosecution’s criminal complaint suggest that Mr. Macy — identified in the complaint as “her spouse” or “Spouse” — was not only aware of the plans, but a participant in at least some of them.

Mr. Singer told law enforcement agents that he met with Ms. Huffman and Mr. Macy at their home in Los Angeles and “explained, in substance, how the college entrance exam scheme worked,” the complaint said. Mr. Singer said he told Ms. Huffman and Mr. Macy that he “controlled” a testing center, and “could arrange for a third party to purport to proctor their daughter’s SAT and secretly correct her answers afterward,” the complaint said, adding that Mr. Singer told investigators that Ms. Huffman and Mr. Macy agreed to the plan.