Does your site have any core extensions installed?

Not asking to blame them, just to see if it a pure core issue or something related to 3rd party extensions (but still might be something to be fixed in the core e.g. in the backwards compatibility plugin).

If you have a copy of that site, could you redo the test with Error Reporting set to Maximum and Debug System switched on in global configuration? IF you can, do you see any error messages when updating from 5.4.0-beta1 to 6.0.0-beta1?

Yes, there are some extensions. Have checked - but seems no one to be a core.
Have re-done it. For me even not really helpful.
joomla_update.php
`#
#
#Date: 2025-08-20 15:46:01 UTC
#Software: Joomla! 5.4.0-beta1 Beta [ Kutegemea ] 19-August-2025 17:00 GMT

#Fields: datetime priority clientip category message
2025-08-20T15:46:01+00:00 INFO 89.245.7.16 update Test logging
2025-08-20T15:46:01+00:00 INFO 89.245.7.16 update Update-Datei hochladen
2025-08-20T15:46:08+00:00 INFO 89.245.7.16 update Datei [ROOT][TMP]/juvm06kchhnb6gdyeD6V9 erfolgreich heruntergeladen.
2025-08-20T15:46:08+00:00 INFO 89.245.7.16 update Neue Version wird jetzt installiert.
`

everything.php
`#
#
#Date: 2025-08-20 15:45:15 UTC
#Software: Joomla! 5.4.0-beta1 Beta [ Kutegemea ] 19-August-2025 17:00 GMT

#Fields: datetime priority clientip category message
2025-08-20T15:45:15+00:00 INFO 89.245.7.16 updater Loading information from update site #3 with name "Joomla! Update Component" and URL https://update.joomla.org/core/extensions/com_joomlaupdate.xml took 0.11 seconds
2025-08-20T15:45:36+00:00 INFO 89.245.7.16 updater Loading information from update site #3 with name "Joomla! Update Component" and URL https://update.joomla.org/core/extensions/com_joomlaupdate.xml took 0.11 seconds
2025-08-20T15:46:01+00:00 INFO 89.245.7.16 update Test logging
2025-08-20T15:46:01+00:00 INFO 89.245.7.16 update Update-Datei hochladen
2025-08-20T15:46:01+00:00 INFO 89.245.7.16 updater Loading information from update site #3 with name "Joomla! Update Component" and URL https://update.joomla.org/core/extensions/com_joomlaupdate.xml took 0.13 seconds
2025-08-20T15:46:08+00:00 INFO 89.245.7.16 update Datei [ROOT][TMP]/juvm06kchhnb6gdyeD6V9 erfolgreich heruntergeladen.
2025-08-20T15:46:08+00:00 INFO 89.245.7.16 update Neue Version wird jetzt installiert.
`

What I have done in addition

• Download the update package again and did a binary compare: It is correct
• Check all core-extensions which are not having the author 'Joomla! Project' of a fresh installation of J5.4-Beta: Only three of them, and those are the same (Editor - CodeMirror / Editor - TinyMCE / PHPass)
• Made a fresh installation of J5.4.0-Beta on my environment and an update to 6.0.0-Beta: Those is working.
• Deactivated all non-core extensions: Update done. Activated all non-core with one step: empty page

Than a step deactivated all non-core, update to 6.0.0-beta, activate each extension step-by-step. Here the extensions which go in failure after switching on:

• System - JCal Pro // permanent error message: Cannot access protected property Joomla\CMS\Application\AdministratorApplication::$input
• System - JCE Pro // empty page
• System - Joomlashack Extension Support // permanent error message: Cannot access protected property Joomla\CMS\Application\AdministratorApplication::$input
• System - Techjoomla Updates // permanent error message: The server returned a "500 - Whoops, looks like something went wrong."

Has taken a longer time. After failure I was not always able to find the entry via SQL to switch the extension off again and had to setup the test-system again...

Cannot access protected property Joomla\CMS\Application\AdministratorApplication::$input

@heelc29 Could that also be caused by PR #45702 ?

@Rolli1962 Just to be sure: Could you test if it also happens without the mentioned 3rd party extensions?

System - JCal Pro // permanent error message: Cannot access protected property Joomla\CMS\Application\AdministratorApplication::$input
System - JCE Pro // empty page
System - Joomlashack Extension Support // permanent error message: Cannot access protected property Joomla\CMS\Application\AdministratorApplication::$input
System - Techjoomla Updates // permanent error message: The server returned a "500 - Whoops, looks like something went wrong."

Already if those 4 entries are deactivated, there are no issues in the back-end. Think there is no need to de-install them - if it is already running if they are deactivated.
Later on I find out, even if in this configuration the back-end is running - but not the front-end. Had to deactivate more. The test-site is currently in this status. Think there will be a lot to do for the programmer of the extensions. Hope they will go into tests...

Think there will be a lot to do for the programmer of the extensions.

@Rolli1962 Seems that they are using code which has been deprecated a long time ago in Joomla 3 and now in Joomla 6 finally gets removed.

The relevant change for the Cannot access protected property Joomla\CMS\Application\AdministratorApplication::$input is documented here for the update from Joomla Framework version 3 (which is used in Joomla 5) to version 4 (which is used in Joomla 6): https://github.com/joomla-framework/application/blob/4.x-dev/docs/v3-to-v4-update.md