Not possible. As some of those links point out, making an APK that automatically installs itself and deletes the original installer without any further user intervention is called malware.
You can write a program that will download and install arbitrary APKs if the user grants it the relevant permissions, but I'm yet to see a good implementation or documentation for this. Typically you can invoke the default system installer using code like this:
File apkFile = new File({path to APK});
Intent intent = new Intent(Intent.ACTION_VIEW);
intent.setDataAndType(Uri.fromFile(apkFile), "application/vnd.android.package-archive");
startActivity(intent);
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