How to change content type name and content type group name

Hello

I manually conetnt type name in one of the site colection, but I noticed it is updating in all the document libraries and each items. Like if change the term name (metadata) and run Taxanomy Updated Scheduler manually, it will automatically update term in each document of every document library.

Is there same kind of job If I change the content type name on web application, so it will update each the content type name in each item and document library?

Using PowerShell script an I change the content type name and content type group name for all the site collection and underneath of each collection document library for a web application using powershell?

Avian