The AUSkey browser extension for Chrome and Firefox does not appear to have any obvious silent install option.

The AUSkeyInstaller.exe uses the Nullsoft Scriptable Install System. Use /S to install silently.

AUSkeyInstaller.exe /S

https://abr.gov.au/AUSkey/Help-and-support/Setting-up-AUSkey-troubleshooting/Download-software-for-AUSkey/
https://auskey.abr.gov.au/plugin/AUSkeyInstaller.exe
http://nsis.sourceforge.net/Docs/Chapter3.html