If you want to add the "send as" permission to distribution group, you can do this:
Use GUI:
1. Went to the mail enabled security group in AD Users and Computers and brought up the properties of the group
2. Click on the security tab and added the security group and provided Send As permissions
OR
We can use the following cmdlet from Exchange Management Shell:
Add-ADPermission UniversalDistributionGroup -ExtendedRights Send-As -user UserWhoSends