Register Internet Explorer (InPrivate) with Default Programs
Download RegisterIEInPrivate.zip, unzip and run the file RegisterIEInPrivate.exe (requires elevation)
Click Register IE (InPrivate) with Default Programs button.
Now you have Internet Explorer (InPrivate) as a separate browser option in Default Programs. Setting this as the default browser would cause URL/HTM/HTML/MHT/MHTML files to open in InPrivate browsing mode by default. (Double-clicking on an URL/HTM/HTML/MHT/MHTML file would launch Internet Explorer in InPrivate browsing mode.)
Editor’s note: If Internet Explorer prompts you with the following dialog, uncheck Always perform this check when starting Internet Explorer, and click No. Otherwise, Internet Explorer (normal mode) would takeover as the default browser.