This is a problem with the Joomla extensions update which is handled by Joomla's own code. When installing an extension (first time or update) Joomla is supposed to look into the XML manifest of the package to find out what is the update information URL. If there is an existing record in the Update Sites management (#__update_sites table) Joomla is supposed to replace it. This is the theory. Sometimes, especially when caching is enabled on the site, Joomla won't do that correctly. As a result the next update will "see" the old update site for the Core version.
Go to Extensions, Manage, Update Sites.
Filter by Admin Tools
in the search box.
Select all entries on the list and click on Delete on the toolbar.
Now go to Components, Admin Tools. This runs a bit of code we put there to make sure the correct update site is enabled.
Reload the page twice.
Now go back to Extensions, Manage, Update Sites. You should see an item for Admin Tools Professional. If you have Joomla 3.9.26 or later you will also see a dlid=YOUR_DOWNLOAD_ID where YOUR_DOWNLOAD_ID is the Download ID you've entered in Admin Tools' component Options page. Future updates will now work correctly.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!