How to configure what members receive private message site errors

Our team receives a private message when there's an error with our site (i.e. Updating the hfbase.less file and there was a small error in the code). This message is sent to a number of team members, but it doesn't appear to have been sent to just users with the Admin Role. 

How can we configure our site error messages to be sent to specific users or users with a specific role?