I suggest you have these users from the organisation in a group. This group should have the following permissions and access
1. They must be SD or JSM users..NB. there is an impact on licensing
2. Have the relevant permissions eg close issue or transition issue permissions
3. Have Service Desk Team role in the related project/ Agent role
You can then further enhance this by using workflow validation on the close transition to ensure that only members in that group are allowed to close the ticket.
A workflow condition can also only allow members of that group to see the close transition.
Using a validation and/or condition all depends on what you want to achieve or prefer.
All right so in that case, I suggest you make use of the New Automation rules so that when the people from the organization comment with a certain key-word eg "Close" then the ticket is automatically transitioned to the Closed status or any status you want. At least this will not require them to have a license.
The rules are very customizable and you can add multiple conditions that must be matched first for the transition is possible.
Please see below the snip I took, NB I was just testing but you can feel free to explore all the possibilities.
Hope this can help.
adding to the perfect suggestions from josephine a similar approach could be to allow the users to execute the transition through the portal:
As it reads it would be ignoring validators you could even forbid your support team to close the issue but the customer would have to do it.
Why I am asking: your requirement can be understood in a way that somebody from customer side must "approve" the request was handled to their satisfaction (^= with close customer approves) OR it could be understood that this should be provided as a nice option but your support team also is entitled to close issues.
The approach with Automation is also very nice and this is what Community is about - to find out a different way to implement something, I really need to look deeper into Automations ;)
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