I am using Activity Starter in my app to open .apk files. I have created a sample app OpenAPK.apk (4.1 MB)
which can run the Package Installer with an APK. Please test it! Just write the path to APK to text box and press button. Like this:
Yeah! It’d be an awesome thing so they wouldn’t have to really do anything, plus it’d be signed by you and it would update it on the Playstore to the latest version too.