Support

I think that the fact that we're having this conversation on a site which runs on Joomla 5.3.1 and has Admin Tools installed answers your question better than anything I could say.

The problem you have is that Joomla, for some reason, has a record in its #__schema table which insists that an older version of Admin Tools was installed on this site when that's not the case. This means that Joomla tries to do an update from that old version it thinks is installed instead of a fresh install. Of course, updating what does not exist cannot possibly happen, hence the problems you have. We will have to convince Joomla that it's wrong.

First, go to System, Manage, Extensions.

Expand the Filter Options.

Under "- Select Type - " choose Package. 

In the Search box type Admin Tools package (important: type the whole thing) and click on the magnifier icon.

Select the only extension you see, making sure it is of type Package, NOT Component.

Click on Actions, Uninstall. Ignore any errors.

Install Admin Tools' ZIP package twice in a row, without uninstalling it before or in between.