They want to save face and avoid looking stupid.

他们想保住面子，避免显得愚蠢。

youdao

These students do not want to get in trouble or seem irresponsible, so they make up an excuse—a lie—to save face.

这些学生不想惹麻烦或显得不负责任，所以他们编造了一个借口——一个谎言——来挽回面子。

youdao

The scandal was hushed up in an effort to save face.

为了不丢人，没有把这一丑闻张扬出去。

《新英汉大辞典》

"What it appears to be is an attempt to save face with the government, which is asking for help,".

VOA: standard.2009.12.03

You try to save face and openly declare that a lost account was too small, anyway?

FORBES: Grace Under Fire

By signing this agreement, Honduras helped Mr. Obama and Secretary of State Hillary Clinton save face.

WSJ: O'Grady: Hillary's Honduran Exit Strategy

The individuals responsible for regrettable decisions may save face but the participants or taxpayers will pay the price.

FORBES: Public Pension Shortfalls and Consultant Conflicts (September 1, 2002 )