I have a total of 5 years with SharePoint development, version varying from 2007 - 2013 projects from multinational companies like Accenture, CGI Group and HSBC.
This can be achieved in 2 ways namely: by feature deployment, which would require you to create a custom look-up site column with a data source pointing to the out-of-the-box (OOB) ID column and attaching it to list A; and through web administration which would only require you to have permission to create site columns.
I hope in hearing from you to further discuss project requirements.