For a possible solution to the dns-issue you might want to look HERE.
You should have given more attention to “Section 1 – Install Debian 12”, step 1.10, step 1.16, and step 1.17.
With this you would have continued “Section 2 - Install OS Agent, Docker and Dependencies” as that non-privileged user you have created with step 1.10 which in turn would most probably have saved you from these kind of privilege issues like “couldn’t be accesses by user '_apt”’ and “(13: Permission denied)”.
The problem shows loud and clear into the eyes:
Maybe you want to re-read and understand the very beginning of the guide?
Armbian is not Debian but a derivative and as such it is not supported by Home Assistant Supervised and neither by this guide!
My two cents for those who get Could not resolve host: error.
Open /etc/sysctl.conf
sudo vim /etc/sysctl.conf
Add these lines to the end of file:
net.ipv6.conf.all.disable_ipv6 = 1
net.ipv6.conf.default.disable_ipv6 = 1
net.ipv6.conf.lo.disable_ipv6 = 1
net.ipv6.conf.tun0.disable_ipv6 = 1
Save and exit vim
Esc + :wq!
Reboot
sudo reboot
After reboot I was able to resolve hosts.
Edited: added sudo as nickrout suggested.
Love vim <3
nano is somewhat easier than vim 
Also worth noting you will have to be root user or use sudo.
I am in the same position:
But the touchscreen is just grey. During boot, text scrolls by, then there is a short colourful display, then it all goes gray. I am currently blocked on this.
I’d like to install X and get the HA dashboard on the screen.
I expect there will be some driver configuration issue, then I should at least get a terminal window on screen. After that the rest should be relatively straightforward.
Looks very much like the supervised installer is blocking the installation on an OS that is not supported. You have installed Debian 12 from Raspberry Pi OS version, did you?
Show us the output of:
cat /etc/issue.net
Quite hard to tell if only some selected and censored log snippets are provided which you personally regard as sufficient to get to the bottom of the issue.
Anyway, first of all:
You should have given more attention to “Section 1 – Install Debian 12”, Step 1.11, Step 1.16, and Step 1.17.
After the OS is installed log out as root and continue installing HA Supervised as the non-privileged user only you have created in Step 1.11. Preferably headless from a remote machine connecting through SSH.
Whenever root privileges are needed prepend sudo before the following command as this is also clearly stated with the guide. Just don’t run the whole install process as superuser (root).
“Sub-process /usr/bin/dpkg returned an error code (1)”":
The above error message indicates a problem with the package installer. This happens after a failed software installation or if the installer becomes corrupted. If the package installer (dpkg) or the package database has become corrupted (damaged) new software installation will cause the above error message.
One way to repair a corrupted package database is to reconfigure it by running the following command:
sudo dpkg --configure -a
If the above doesn’t help you can try to to fix the dependencies in the package installer by executing:
sudo apt-get install -f
where the -f option means fix-broken. The above command will fix any broken dependencies in the package manager which occur when a download is interrupted or there is a problem with the cached download.
If you have no clue about what I try to explain with the above this installation method is certainly not for you and you’d better off installing HA OS. This thread is not about providing private Linux lessons. If you really want HA Supervised then follow the guide strictly and by the book and you should be good. Please refrain from throwing screenshots of some log snippets into the forum and expect the community to solve the issues you ran into because you did not follow this guide strictly.
I had this error also but it disappeared after rebooting
I have been running in too many issues lately (mostly because I also run docker-compose to run web servers) and my Supervised Home Assistant setup keeps breaking. Therefore I decided to just install Proxmox on top of my Debian installation.
Does anyone how how to cleanly undo the Supervised Home Assistant install (after I moved to a Proxmox VM of course 
)?
Reverse the steps in post 1.
Why ? I used to always install under the root user. However, then errors usually occurred with an enviable frequency.
because it’s really bad practice to run as root user and as you see it screws things up
Is it possible that a recent docker upgrade messes things up again?
Upgraded docker-ce as part of an overall update of the underlying Debian system earlier on, and now supervisor won’t set up anymore.
To verify, I did the same on a backup system I have and the exact same thing happened.
Reading through this thread and some others, it appears that similar things have happened in the past…
Running suoervised on Debian Bookworm on an Intel nuc 8.
Just had the exact same issue myself. Not yet figured out how to fix
Clear. I checked, there wasn’t a bug reported yet as far as I can see.
Now there is.
Someone posted this as a temporary workaround, basically a downgrade of docker ce
sudo apt install \
docker-compose-plugin=2.21.0-1~debian.12~bookworm \
docker-ce-cli=5:24.0.7-1~debian.12~bookworm \
docker-buildx-plugin=0.11.2-1~debian.12~bookworm \
docker-ce=5:24.0.7-1~debian.12~bookworm \
docker-ce-rootless-extras=5:24.0.7-1~debian.12~bookworm