I agree but the House Oversight Committee says she waived her right to claim the fifth back in May when she swore under oath she did nothing wrong.
If she is a defendant charged with a crime, then the Fifth applies to the entire testimony, where you can't pick and choose which questions to answer. If you start answering and then suddenly plead the Fifth, you can't do that. You don't get to tell you side of the story and then plead the Fifth when it comes to cross examination about those statements.
However, if you are a witness, or a suspect being questioned, especially if you are testifying because you were subpoenaed, you can pick and choose which questions you answer. The only way a witness can be compelled to testify is if they have been given immunity from prosecution for anything the witness states.
If the Oversight Committee actually charged her with a crime, and she answered any questions at all under oath, then she can't plead the Fifth. But if the questions were part of the ongoing investigation, she can. In this case she made a statement, not in response to a question, and then promptly pleaded the Fifth. The Committee voted and deemed that she waived her right to plead the Fifth. But if that goes before a court they'll likely lose that argument.
The tricky part is how to characterize Lerner’s testimony before she invoked the Fifth Amendment. Was she simply expressing her opinion that she is innocent? Or was she asserting actual facts about a matter that could subject her to criminal prosecution by the Justice Department?That difference is the key to a court’s eventual determination of whether Lerner waived her Fifth Amendment rights. Her short statement is open to interpretation on that issue. If she was asserting facts, then she's subject to cross examination of those facts. Because it's open to interpretation, it's highly unlikely that a court would rule against her, especially since she was forced to be there and wasn't yet asked any questions.