Can't invite non iCloud calendar (Google Calendar) to iCloud calendar entry

I have a iCloud Calendar, a private Google Account and a family Google Account.

(Everything is done in the Apple Calendar App)

When I add a new entry into the calendar on my family Google Account, and invite my private Google Account, I get a notification in the Calendar App where I can accept the invitation. Also, I get the invitation in my Google Calendar App, where I can accept it as well. So far, so good.

But now when I make the entry in on my iCloud Calendar Account and invite my private Google Account, I don't get any notification and in my Google Calendar App I don't see anything.

I don't use the Gmail Address as my Apple ID!

How can I manage to invite a non iCloud calendar Account to a iCloud Entry?

If you need more information, feel free to ask. :)