It seems to me that this would almost be core functionality in Joomla, but it's not. Having a single homepage can detract from a user's experience. So today I began using the "Nomad Plugin" for Joomla on one of my sites. I use the Purity III template by JoomlArt (this template) almost exclusively now even though it has a whopper of a learning curve, but once you know how to properly manipulate the design, you can accomplish some pretty amazing things. When Purity III is in mobile mode, it displays the sites logo or name as a button and I need the target of that button to change depending on if you are a guest, or an authenticated user.
A couple of notes on Nomad - first, it does what it says -
Nomad is not login redirection, it's homepage redirection!!! Joomla gives you a single homepage - Nomad gives as many as you need! Per-group or per-user - as many as you need.
but, there are some caveats.
First, it does not override the default joomla menu item - where you put the star still determines what pages new visitors will see. Second it does redirect users once they are authenticated (social, joomla or EasySocial) logins. Once the user is identified as a group member - Nomad will silently redirect that group to the page you have assigned with almost no overhead.
If you are using Nomad or trying to accomplish the same thing, I would love to hear from you in the comments. Also, please excuse the emptiness of the site, recently migrated to 3.2 and still moving data over, so please be patient.