by Wayne Smith
Google and other search engines are generally very good at parsing pages on a site to determine the site name. But on occasion the site name may be ambiguous;
In some cases, a site name shown for a home page may not have propagated to appear for other pages on that site. For example, example.com might be showing a site name that's different from example.com/internal-page.html.
We're actively working to address this. We will update this help page when this issue is resolved. In the meantime, if your home page is showing the site name you prefer, understand that it should also appear for your internal pages eventually.
Schema can be used to make this information unambiguous.
The schema should be on the home page for the domain. In this case on the content that appears when somebody goes to the URL https://www.example.com/ -- The home page needs to be readable by Googlebot.
It has long been a best practice that links pointing to the home page use the site name or brand name of the site as the text, not the text "home", and that the site name or brand is part of the title of the home page.
Using isPartOf on internal page
The isPartOf property can be used on internal pages to provide the information as to what website the internal page should be associated with. Although this method is not specified by Google. And, having different names for isPartOf may lead to the situation of the site name being ambiguous. It is noted here as a potential way to resolve the problem and as something that needs to be specifically checked to make certain that there is no disagreements being provided to search engines as to the site name.