If you choose to forward your email from Google Groups there is an additional forwarding step required. Google requires this as a security measure to ensure authenticity of the account.

Once you have your forwarding address from the Groove settings, go into your Gmail account under Settings > Forwarding and POP/IMAP and click the button to Add a forwarding address. 

Next you'll need to paste your forwarding address into the popup modal and follow the steps to send a confirmation email to your forwarding address.

When you receive the confirmation email in your Groove account, click on the link in the email to confirm your forwarding address. 

After you've confirmed everything, you are good to go!