Create a user account called forwarding-user@yourdomain.com. Then make forwarding-user@yourdomain.com a member of your group group@yourdomain.com. Then for each of the individuals outsider@somedomain.com who need to email group@yourdomain.com create a filter in forwarding-user@yourdomain.com to foward their mail.
When outsider@somedomain.com emails to forwarding-user@yourdomain.com, it is forwarded to group@yourdomain.com. Messages in group@yourdomain.com go to forwarding-user@yourdomain.com but go no further. For cleanliness, you could create another filter that deleted all the messages from group@yourdomain.com.
Of course, this would be very cumbersome if you have a lot of outsider@somedomain.com users.
I can create an email address like myname@companyname.com ...
You don't necessarily have to use Google Apps (as Prasad states) in order to send email using your Gmail account from a particular email address that you already own and are currently able to receive email to (and preferably send email from also).
It should be noted that by default, it will use Gmail's SMTP server in order to send email. In this case your Gmail address is still present in the Sender header, so depending on the recipients email client, your recipient might see something like:
From: myname@companyname.com sent via yourgmailaddress@gmail.com
Obviously this isn't necessarily appropriate as a business solution. However, you can use your own SMTP server that is associated with your domain. This will remove your Gmail address from the sent email and so the email recipient should be none the wiser - it will look as if the email has been sent from your actual email address (no "via" notice), because it has been sent from your actual email address using your own SMTP server. When setting up the account in Gmail (see below) you must explicitly pick the second option "Send through yourdomain.com SMTP servers" and enter your details.
In Gmail Settings > Accounts and Import > "Send mail as:", click "Add another email address you own" and follow the instructions. You will be sent an email to this address with a link that you need to acknowledge to prove that you own this email address. You can choose to send mail via your SMTP server, or use Googles (the default).
In order to receive this email to your Gmail account you can either set up a forward at your current host, or set up Gmail to collect a mailbox using POP3. There is a limit of 5 POP3 accounts you can setup in this way.
This is also a great way to filter out spam if that is a problem - Gmail is a very good spam filter!
However, as mentioned, this isn't necessarily a complete solution - this uses your Gmail account to send/receive email from your server. Your server is still receiving the email. Google Apps allows Google to manage everything.
Best Answer
Definitely the new the collaborative inbox from Google Groups for Business if you are on a Google Apps for business.
If not it worth the upgrade.
http://support.google.com/a/bin/answer.py?hl=en&hlrm=en&answer=167430