How can I download the macOS Big Sur installer on a Mac which is already running Big Sur?
You can download the official macOS Big Sur installer from Apple  by using the following steps:

On your Mac, open the Mac App Store app, search for Big Sur and click on the Big Sur app to access the macOS Big Sur app page.
You can also use this link to directly access the macOS Big Sur app on the Mac Appe Store:

https://apps.apple.com/app/macos-big-sur/id1526878132

If the App Store app doesn't launch automatically, you can click on the View in Mac App Store button on the above webpage.


Click on the GET button.


Clicking the GET button would automatically open the Software Update preference pane under System Preferences app.

Click on the Download button to download the macOS Big Sur installer.


The Software Update preference pane would start downloading the installer. You can view the download progress here. If you wish to cancel the download, you can do so by clicking on the circular button with a cross shown just ahead of the progress bar.


One the installer is downloaded, the same can be found under the /Applications folder with the name Install macOS Big Sur.app.