• update app from within another app?
• another option
• https://github.com/NDMAC/apk-updater
• https://github.com/feicien/android-auto-update
• 2018 blog about it (Requires at least API 21)
• Update Checker for API 8
• install without reboot by signing and zipaligning
• silent installs by calling shell commands

Starting with API 21 it's possible to have a managed device mode which will allow for silent updates without root. Before this the app needs elevated privs.