Below are steps so that, from your personal email account, you can "send as" from the same email address as that used in a Google Group under a business account. The information is not really spelled out anywhere, so I'm writing this down for future reference.
1. Login to the business account primary email (the owner / admin).
2. From Gmail, go to Settings > Accounts and Import > Send mail as. Add the Google Group address here once.
3. Login to your personal email account in Gmail, which should also be a member of the Google Group to receive messages.
4. Go to Settings > Accounts and Import > Send mail as. Add the same Google Group address here again.
For the login credentials, you'll need to use a user email (non-Google group) from the domain of the business and for password, it needs to be an app password generated specifically for this.