Home assistant supervisor stuck at installing status, even after I restart the home assistant.
Restart or Reboot? Reboot(power loss) may be needed
If you can access via terminal, try
ha supervisor restart
ha host reboot
I had the same issue (with the same versions) after a reboot of my HA today.
Problem solved with ha supervisor restart. After this, supervisor told to be 2026.06.1
Then use this 
in Terminal or Reboot
ha supervisor update
Tried restart then supervisor become unavailable. Unplugged NUC now seems everything working. Feels like there's a bug with this new update.
Leave it alone & don't restart or attempt to update it manually.
Supervisor is the only thing in HA which updates automatically. There were probably too many requests going to github at the time you tried, so it stalled.
Restarting when any update is in progress is a surefire way to mess things up, so just leave it alone & check on it the next day - it'll eventually update by itself.