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You can do that by creating a dedicated Automation for Jira rule on your project settings page.
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First, add the ‘Issue commented’ trigger and as comment type select 'Comment is the main action'.
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Next, add a new condition component ‘New condition’ -> ‘Advanced compare condition' to verify that the comment was added by the External Share for the Jira user.
Set:
First value: ‘{{comment.author.accountId}}’
condition ‘equals’
Second value '5caf360937b1dc049479af84’
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In this example, we will change Issue priority to High by adding a comment body that will trigger this action. You can use for example priority=high comment body text.
Add new condition add a new component ‘New condition’ -> ‘Advanced compare condition'.
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First value: ‘{{comment.body}}’
condition ‘contains’
Second value 'priority=high’
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Next, add the new action a ‘New action’ -> ‘Edit Issue', choose the 'Priority’ field and select its value to High.
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That’s all, now comment with content ‘priority=high’ added on the shared page will change issue priority to high. To see changes on the shared page you need to reload it.
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If you also want to allow an external user to change the priority to other values, you need to add separate automation rules to that.