would be grateful for an explanation.
Which version of Windows is installed on the machine.... Are we talking about a Windows Pro machine or a Windows Home machine.
The problem, most likely, is a group membership conflict.
IF you go back to the Properties -> Security tab as pictured above
You'll notice that in addition to the Administrators Group there is also the Users Group as well as Authenticated Users either of which may have more restrictive permissions
AND
On an NTFS volume the most restrictive permissions win every time.
On a Pro machine I generally just launch
lusrmgr.msc
And make sure that a computer Administrator (member of the local administrators group) has only that group (Administrators) that they are a member of.
That particular Snap-in Console is not available on a Home version computer, so....
You'll need to select either OR if present, both "Authenticated Users" and "Users" and check the box in the bottom pain next to (at least) "Modify" (which means they can do anything with the file except delete it OR "Full Control" which includes "Modify" permissions as well as the ability to delete the file.