During crisis: remain level-headed and communicate as much as possible
During a crisis, pressure on the organisation and on individuals mounts very sharply. Cheap advice is to put your CEO in the spotlights, to "show leadership and empathy", but as an automatic response this can prove to be a costly mistake. Once the CEO has been fielded, there is no escalation room left available. The CEO is also from that point forward inextricably associated with the crisis and therefore damaged.
While you assess the depth and breadth of a crisis, use an authority ladder to protect your CEO as long as you can. The escalation runs from media relations spokesperson, via experts, managers, senior managers, to board members and eventually the CEO.
Finally, any crisis that actually merits fielding your CEO will inevitably need not just empathy, but the start of a complete process of repairing relations and preventing recurrence.