GJ Message Members Error

When using the message member function, after clicking Save, the page just hangs for a few minutes and then produces a 500 error:
The server encountered an internal error or misconfiguration and was unable to complete your request.

Please contact the server administrator, webmaster@domain.com and inform them of the time the error occurred, and anything you might have done that may have caused the error.

More information about this error may be available in the server error log.

Additionally, a 404 Not Found error was encountered while trying to use an ErrorDocument to handle the request.

The error log says:
Script timed out before returning headers: index.php, referer: www.domain.com/index.php?option=com_comprofiler&task=pluginclass&plugin=cbgroupjive&action=groups&func=message&cat=1&grp=2&Itemid=100114

Any thoughts?

Enable debug mode and maximum error reporting in Joomla global configuration. Once done try to use it again and provide any fatal errors display.

I don't see any fatal errors appear. All the text on the page just before the 500 error appears is attached with private info redacted.

Also, I neglected to mention in the first post that the messaging process actually completes and all group members are receiving the message. The problem is that the Message Members page doesn't go away so the group owners are clicking "Save" repeatedly thinking that it is not completed. Doing this causes multiple messages to be sent.

Please check server error log and see if there are any fatal errors present there. If none then please PM backend super administrator login credentials and will take a look.

After further testing, the process will complete for small groups, although very slow. E.G. for a 13-member group the process is taking about 50 seconds with Notifications set to "PMS and Email". Where we are seeing the time-out error is in larger groups. So maybe we can resolve it by increasing the timeout settings on the server, but the process length will still be a problem, because users think the button didn't work (or something) and will click Save again and again, sending multiple messages to members. Here are test results for the process completion time of a thirteen-member group:

PMS or E-mail: 13 - 30sec
PMS and email: 45 - 50 sec
PMS Only: 12 - 30sec
Email Only 30 - 50sec

Do you think this time duration is normal? We have notifications currently set to PM & email, so it's worst case.

Sending 45 emails and 45 private messages all at once is going to be pretty slow. You'll need to increase server memory limit to at least 124mb, increase timeout time to at least 60 seconds, and just wait for it to complete. Alternative is to use PMS Only, Email Only, or PMS or Email.

It's hard to say what is normal. It depends on your server and your site. If you site is in general slow then it's going to get even slower using such a heavy feature (which is why only admins/owners can use it).