Google-workspace – Use the own domain as a google sites URL with Google Apps

I have several Google Sites at my organization that we use for different projects and audiences.

I'm already using the feature that lets me map a subdomain of my domain to the google sites by adding a CNAME record of "ghs.google.com" — for example, I have project1.example.com which redirects to https://sites.google.com/a/example.com/project1/Home. After the initial redirect, though, all of the URLS are at google's domain:

• sites.google.com/a/example.com/project1/Home
• sites.google.com/a/example.com/project1/AnotherPage
• sites.google.com/a/example.com/project1/AnotherPage/Subpage

I would like the visible urls to be:

• project1.example.com/Home
• project1.example.com/AnotherPage
• project1.example.com/AnotherPage/Subpage

I'm not concerned about http vs. https.

Is there a way to do this using Google Apps?

UPDATE (10/26)

I think the problem is that this is a private google site and I require a login for people to view it. When I temporarily published it publicly and visited without logging in, the urls mapped and displayed as I wanted. For privacy reasons, though, this site has to be password-protected. Do you know of any way to retain the subdomain mapping for logged-in users?