It is an interesting question.
They say that goods once damaged can be fixed but they never look like new.
Reputations are the same after a betrayal of trust. Every time the coverup is repeated, the pain is repeated as people feel cheated. It cannot be good for the staff either who may well have devoted many years of their lives to a company and see its culture is loose with the truth.
The only way out, is for the company to restore its reputation is by being honest. Focusing on the roots of the problem, then asking itself why did it lie and not invent more coverup. That way its employees and customers can believe again. A people have long memories, it may take time to forgive and forget.
I worked for a charity organisation where an official was found stealing some of the fund collected. Of course, we were upset; we had assured people that their monies would be properly looked after, and we were caught this official lie. What did we do?
We went to all the big contributors and held public meetings for those that had provided. We told them the facts and invited them to see what we did differently. How we intended to make sure it never happened again. As people were impressed with our honesty, we quickly made up much of the damage.
I do believe ultimately it is your actions and behaviour, that make the difference.