Using a group email address for multiple signers to pick up and sign agreements

We send out thousands of agreements that need to be signed. We would like to give a group of people the ability to sign these documents. So, we'd like the signer email to be a group email address that forwards the agreement to all the members of the group. Then, anyone in that group could go in and sign that agreement. 

If we did that, would the name associated with the group email just keep changing? What if they assigned it to their own email address, would the confirmation still go back to the group email address?

Is there another way that we should be doing this?