Hello! There is an admin group and a Better Chat plugin, I registered myself in the admin group and deleted it from the default group, but after each restart, the default group is issued back. How do I make it not automatically go into it?

All playeres are automatically in the default group in Oxide, that isn't from Better Chat. There is no way to disable this natively without writing a plugin to remove that group (and potentially break functionality). The admin group is also from Oxide, not Better Chat, and players are assigned to that automatically based on the auth level 2 being granted in Rust. Both of these groups can exist normally, and are not meant to be deleted.

If you want to hide a chat tag or something, Better Chat has options for that, which you can read about on the plugin page and the many existing threads that answer this.