The process for setting up an 'Email' action on a Case varies, depending on when your organization was created, and whether Email-to-Case feature is enabled.
Create a Case specific action of 'Send Email' type, using instructions on Create Object-Specific Quick Actions
Note: If you are still using the Classic UI and Enable Case Feed Actions and Feed Items , then add the Email action only to the 'Mobile & Lightning Actions' of the Case page layout, it will not appear in Salesforce Classic full site. To ensure it also shows in Classic, also add to the 'Quick Actions' of the Case page layout.
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