You could take a look at SquirrelMail (simple but does its work), Roundcube (nicer) and Horde (this has more muscle, so to speak).
SquirrelMail is a comparable solution to Roundcube.
After looking into just what Roundcube offers though, it seems to be the best things around for small-to-huge organizations not using something along the lines of Office 365 / GMail.
The email protocols used are the industry standard and are the same type of email addresses someone would have "@CompanyName.com" as an employee of a company for example.
The code is open source and I do NOT have access to emails once users change their password.
Yes it is.
You can choose from 3 different ones too.
You can see the protocols here:
Quite secure!
It's the same standard web email that an employee would have if they worked for a company and had an email address "@CompanyName.com" for example. I do NOT have access to any emails.
You can see the protocols here: