The following message appears at the end of JRE online installation(after closing installer dialog):
Abort - Java(TM) Installer
To restart the Java(TM) installer, please refresh the web page.

JRE installation is OK, but there is no AU installed(No Update tab in JCP, no AU registry entries...)

There is no message at the end of offline bundle installation, but AU does not install and MSI log says:
Error 1406. Could not write value SunJavaUpdateSched to key \SOFTWARE\Microsoft\Windows\CurrentVersion\Run. System error . Verify that you have sufficient access to that key, or contact your support personnel.

Logs attached.