Internet Explorer disables all browser extensions when entering private browsing mode which means that:
- Your extension never runs on tabs in private browsing mode.
- Instances of your extension running in other tabs are neither aware of nor can they interact with pages in private browsing mode.
The unfortunate result of this decision by Microsoft is that even if you had a method of detecting incognito status on a window [1] you wouldn't be able to do anything with that information!
[1] e.g. Writing a BHO that enumerates all desktop windows and checks for the string "InPrivate" in their titles.