Cameron Herold explains that a critical skill for COOs is the ability to say no to the CEO in a protective, discreet way—acting as the 'brakes' to the CEO's impulses and preserving the CEO's ego and brand. He emphasizes timing (away from the leadership team or advisors) and framing (not as argument but as saving them from themselves).
Chapters
One of the most important skills that the CEO need to be great at, that pretty much no one else in the company needs to be as good at, is you need to be able to say no to the CEO and you need to be able to save the CEO from themselves and be able to do it in a way that you protect the ego and the fragile brand of the CEO themselves. An example is the COO needs to be the brakes to the CEO's gas. Or as one of our CEO alliance members said, the CEO needs to be the leash the to her dragon. The skill that the CEO needs to be really good at is to be able to say no to them. Or not now, but do it in a place where the leadership team isn't around, the other team members aren't hearing. You're not doing it in front of the Board of Advisors, and you're doing it in a way that the CEO understands that you're not just arguing with them or debating them on everything. You're actually trying to save them from themselves. It's a core skill that every COO needs to be great at.