The ones to dominate in groups tend to have social engineering skills, or "Ben Linus skills" you could call them.
They don't need to know how to start a fire, or fire a gun, or fix a car. They can just get other people to do it for them -- not for money, but because they can con them in to it, or exploit their weaknesses, set them against one another, or bribe them with promises.
This is a problem is too small group in a survival situation. A viable group should have the size of a village at least - over 100 people of some diversity.
Otherwise it is very fragile.
Obviously enough resources to start with.
You have a SERIOUSLY optimistic view of the Mad Max type aftermath of a nuclear apocalypse. Fix a car? Land a plane? Those questions imply an assumption on your part that an awful lot of support systems will have survived. (For instance, I'm not at all sure that fuel deliveries would continue to be made after a nuclear apocalypse. I'm even less certain that the electronic systems of most planes and autos will survive the nuclear unpleasantness unaffected.)
Indeed the main item as usual is how to get your army to listen to you. A hard people skill. Logistics is always useful as well.
A trade or skill is a good fallback in case of exile or loss of resources.
Setting aside how loaded that statement sounds, the role in question could easily involve espionage and assassination.
What are your Mad Max skills? If the nuclear apocalypse comes and you survive, what useful skill do you have that will let you continue to live without the rule of law? Can you build a house? Fix a car? Grow crops? Butcher an animal properly? Land an airplane? I've always though everyone should have at least a few of these practical hands-on skills, in case the shit hits the fan. What good are your computer programming skills when there's no more electricity, let alone functional computers? So, you're a social media marketing manager for a global brand? If that's all you know how to do, you'll end up some local warlord's sex slave.
Why would "security forces" listen to some formerly rich corporate CEO? What has he to offer them?