I won’t take credit for this idea. It was posted on r/worldbuilding but was nuked by the mods for “asking for content”, as usual only after lots of interesting discussion.

OP: "I consent"

Other worldbuilders: "I consent"

The mod: "I don't"

My first thought is that such a scenario would lead to a rise of fascist governments as nations urge to prepare to fight back. These fascists invade their neighbors in hope that they could gather an equal army... somehow.

When 6 months pass and no aliens show up, the fascists whose entire existence was justified by this imminent threat start spreading disinformation about the aliens' message to ensure that their populace still believe in the promised invasion.

Eventually, the fascists exhaust their reserves (and their lies) and the War ends. The broken moon is forever associated with fascism and a conspiracy theory spreads that claims that the fascists themselves crippled our moon.

Not really a rebuild, as it was just the Moon. But where did the debris go? Still in the same rough area/orbit? Maybe that rained down and caused destruction that the rebuilding is for? Did it affect the Moon's tidal effect much (not if it's in the same area maybe)?

Then the question of why they didn't come back.

Mistranslation? They're coming back probably, but the months might have been Earth years or more?

Something worse got them, and potentially knows now about us in some way (dark forest issue)?

It was a test to spark us in progress, either to ready us for something the aliens actually need help with, or to see if we're a danger at all to them with the right motivation (a bit of dark forest there too). Or maybe the only way a species can break through the Great Filter and not kill themselves is to be united by a common threat, and once past that point they survive much better, so they're nurturing in an odd way.

Something else entirely, where we think we're that important to come back to, but the aliens were distracted by far bigger things, not just another enemy but some other threat to them. Maybe a full tech/physics failure, i.e. like in Star Trek Discovery with warp drives, and what was terrible for warp-capable species doesn't affect us since we weren't at that level, so we find other methods and end up being dominate.

So, first, you have the question of what humanity does. You'd need a plan on how to respond immediately in order to keep some form of world order. You'll likely see a division between two groups of people, those who believe they are working towards survival and those who don't.

I don't expect a unified response to survival, but I expect they will probably try to not stop on each other's toes unless absolutely necessary. These are going to be the nations that raise armies and the manufacturing base necessary to go to war or flee.

To those populations who don't see hope, I expect a rapid decent into chaos. Social order collapses and it is likely that those with a survival mentality have to soft land the chaos parts of the world. This likely includes raiding in order to take equipment and shut down dangerous facilities. Any defenders likely get co-opted by the survivor group.

At six months, it is likely that billions are dead in the chaos zones as their ability to sustain their populations collapse. The survivor groups are likely highly militarized and poised for a response that isn't needed. I expect, after some debate, that the survivor groups further expand into the chaos lands for resources to feed their machine. The machine is still kept ready for a few years, a debate occurring as to the necessary readiness but still considered a necessity for the next few years.

OP, this reminded me of an absolutely stunning Twilight Zone episode, not exactly your premise, but it's worth a watch:

https://www.youtube.com/watch?v=fbT1fCHOjfI