A non-report, but SOQL solution would look like this:
SELECT count(id), createdBy.name
WHERE createdDate = THIS_WEEK and ParentId != null
GROUP BY Createdby.name
If your outbound emails are logged as Activities, then you can use Report Type Activities with Cases, Filter by Activity.Type and Group by Activity Created By ...
I just had to do this for one of my clients. The solution is not intuitive at all. What you have to do is create a separate bot to serve as your "language router". Connect that bot to the button config that's used in your deployment (classic deployment or snap-ins/embedded service).
In the bot, you have to read in the data from pre-chat. You can do ...