Answer:

The ability to copy a report is not controlled by a capability.  It's controlled by whether or not the user that copies the report has permission to modify the report.

If the user has permission to modify the original report (that they chose to copy) they'll see the Save buttons.  If not the report will open in read-only mode and the report has to be discarded.

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Potential Solution:

Background Information: You can't grant an individual User or User Group modify access to an existing report.  The Permissions of the Report are likely to be set as the default Sharing Policy: "Private" and Mandatory Permission: "Minimum Access".  The screenshot below shows the Minimum Access Mandatory Permission screen.  It shows that all members of the System Admin User Group possess the ability to Read, Modify and Delete Any Item that has the Minimum Access Mandatory Permission.  Users that are members of the Everyone User Group (all users are permanent members of this group) have access to Read Items they own.

1) For an existing report that you want a user to be able to copy:

Temporarily add the user to the System Admin Group - explained here.  They'll then be able to copy the report (if it has the default permissions - explained above).

NOTE: Membership of the System Admin User Group provides a massive amount of capabilities and permissions.

2) For all reports created in future:

Configure relevant Permissions:

1) Create a new User Group e.g. "Report Administrators".

2) Add any users that you want to be able to copy reports to the User Group.

3) Create a new Mandatory Ruleset e.g. "Customised Reporting Ruleset" that provides the Report Administrators User Group Full Access.

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4) Change the Mandatory Permission Ruleset for Reports from Minimum Access to Customised Reporting Ruleset

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5) Reports created from this point forwards can be edited and copied by their Owner (almost always the person that created the original report), or members of the System Admin or Report Administrators User Groups.