How do document permissions differ from its document library and a content roll-up webpart?

Question

I have an excel spreadsheet (a group schedule), in a document folder, on a page that only IT management can see/access/edit, in a site that all IT group members have access to.
     for example: http://IT Group site/IT Management Page/Schedule/2015.xlsx
On the group's homepage, I have a content roll-up web part that displays all document library contents in the group site.
IT management consists of 3 people. They can all edit the spreadsheet from the original location, but when they access the document from the content roll-up web part on the group's homepage, they cannot save any changes.
     Accessing the file from here: http://IT Group site - content roll-up web part link "2015.xlsx" points to "IT Group site/IT Management Page/Schedule/2015.xlsx"
The error received says "'http://IT Group site/IT Management Page/Schedule/2015.xlsx' is read-only. To save a copy, click OK, then give the workbook a new name in the Save As dialog box."
Where would the permission issue be?
As the SharePoint site owner, I can access the file without any problems saving from either location. Also, I created an IT Management group that has full control over the IT Group site/IT Management Page, and full control over the IT Group site.
Thanks for any info you can share.
Stephanie

Answer 1

Thanks Cameron -
I have no special settings on the document library where the schedule is saved. No content approval necessary or different permissions. The library is set to open documents in client application, and check-in, check out is not activated.
I have logged in to one of their computers. They can access the spreadsheet directly from the document library and make changes freely, and save that version directly to the server, but not from any web parts. Just to test, I created a summary link web part
on the same page that lists the document library, created a link to the file, and the file opens, but they cannot save any changes without renaming the file. The only place they can truly edit the spreadsheet without renaming is from the document library
directly.