Hello Jack,
As you mentioned, you need to add the users who are supposed to be the support chat engineers the role (not group) called “Chat support engineers“. I assume you are using Kentico 8. To do so, go to the Users application, pick a user you want to be the support engineer and edit them. While editing the user, go to the Roles tab, make sure the site selector is set to option (global) and click the Add roles button. Unless you previously made any changes in default global roles, you will see the “Chat support engineers” role. All you have to do is select it and save it.
Another approach is editing a role the user is already a member of and adding the permission for support chat. Go to the Roles application, pick the role you want to edit and see the Permissions tab in the editing interface. Select Permission for: Module – Chat. Now tick the checkbox on the row Enter support. All members of the edited role will now be able to see the support chat in the header.
From your question I cannot say whether you have the support chat enabled. If you need help in this matter, see the https://docs.kentico.com/display/K81/Using+support+chat documentation (it is similar for version 8 and 8.1).
If the support engineer is not able to see the support chat button in the administration interface immediately after properly configuring their user account, try clearing their browser cache (by pressing CTRL+F5) or try opening the administration interface in a different browser where the page has not been visited before.