Access to clean water seems obvious. The goals (motivation, actions etc.) of a person or group) would seem to be a writing choice rather than a worldbuilding problem for us to solve. Can you clarify.

The main background feels like it could be trimmed, while the restatement of question could be broken up to be more digestible. I'll suggest another edit.

After suggesting said edit, said question might be slightly too story-based, as it could heavily depend on their environment and quickly become an infinite list of possible threats to survival. I'd wait for more seasoned users to make that call

@TheCommander I'll introduce some more conditions.

You may also want to choose a specific apocalypse; circumstances differ wildly between them. Barricading for TWD-style zombie hordes, safe food in the case of nuclear fallout-both examples of important goals in their settings

The leadership can, and must, be able to carry out every part of the objective (e.g., all aspects can be centrally controlled by a group of 6-7 people and authority or sovereignty not need be subcontracted outwards) seems like it might be a problem. What if your group needs to build something?

@TheCommander you could have one of the group oversee the construction effort. What I mean is that there needs to be centralized governance in the same sense that en.wikipedia.org/wiki/Officers_of_the_United_States exist. Only 6-7 people with such authority should exist within this group.

Let us continue this discussion in chat.

This is essentially the plot of the excellent, well-researched, and historically significant novel Alas, Babylon (Pat Frank, 1959).

This question is on the verge of being story based. You say they do not know if anyone else survived. They don't know if defenses are needed, since they don't know what to defend against. That means building defenses or not will depend on character motivations.

"A small group of survivors": It depends of how large this small group is. Political stuff doesn't really come into play unless the group has at least a thousand people or so, maybe a little less, maybe a little more; for smaller groups, having any kind of political structure more complicated than John is the boss and Mike is his deputy would be a stupid waste of resources. As for building anti-missile defense systems, the small group needs to have millions of people to even consider it.

Politics happens in any size of group, whether between prisoners, hunter gatherers or whole countries. There are definitely different solutions for different sizes of group. 'John is the boss' (sounds great!) doesn't seem to be a model that emerges in small groups and, unless John is a saint, will require enforcement. Knowing the size of group is important though for systems that might operate successfully.

I won't comment on the political part but as for ideas of possible objectives, I'd recommend this book: en.wikipedia.org/wiki/…

This is quite an interesting article/book on billionaires planning for an apocalpyse and trying to figure how to use their current power to ensure power post apocalypse. It's not enormously optimistic in outlook, but interesting the things considered. onezero.medium.com/survival-of-the-richest-9ef6cddd0cc1

@AlexP "John is the boss and Mike is his deputy" is politics, because they've got to stay in power while balancing competing interests.

VTC:Needs More Clarity. The needs of 20 survivors is a LOT different than the needs of 2,000 survivors. The needs of someone in a temperate environment is a LOT different than the needs for an arctic environement. Are they subject to acid rain? Continuing fallout? Nuclear winter? Are we 5 years after the event, or 500? Are we still dealing with cancer? Is the food supply still dealing with mutation? When you say "missile defence" (hah...), is another group nearby actually organized enough to have airplanes, but your people don't? This sounds a LOT like a storybuilding question.

In fact, if I could change my vote to Too Story-Based, I would. The more I think about your question, the less it's a worldbuilding question. Worldbuilding is independent of all stories. Worst of all is the statement from the help center, If ... you aren’t sure what a character (be it an individual or organization) should do, that is out of scope for the site. You're literally asking us to make a decision for your survivors, which is prohibited.

"what is the most important political or economic objective" - According to which ideology? Communists and capitalists won't have the same priorities. Nor would libertarians, isolationists, vegans, etc.

"Is it to construct shelter" - it depends on how well-suited their current shelter is. "Would it to be to construct defenses, such as anti-missile systems" - it depends on whether they're expecting missile attacks (I don't think I've ever seen this in post-apocalyptic fiction, because missiles are hard to come by, they're associated with advanced societies, which doesn't fit the typical setting, and blowing stuff up with missiles doesn't usually help the typical post-apoc goals of survival and scavenging). Basic governance would exist from the start, and evolve slowly over time.

All this discussion when the obvious answer, as to all questions like this, is ofcourse a Bath Towel..

Given that this is in chat now, I'll add that the point of my comment above was to highlight that "it depends". The most important objective depends on the circumstances and world they find themselves in (if you're telling a story, this is up to you). But the Hierarchy of Needs (as an answer mentions) gives a good framework for possible objectives.