The "Offline Global Address List" is used by Outlook to perform queries against the Global Address List without introducing additional strain on the Exchange environment. While this Offline Address Book is automatically updated every 24 hours, it can be manually updated from inside of Outlook.
In this scenario, an on-premise active directory user was created and synced with Microsoft 365. The Microsoft 365 user has an Exchange Online mailbox, but this mailbox is not visible in the on-premise Exchange Admin Center.
While the Microsoft 365 admin center GUIs are decent, there are many Microsoft 365 management tasks that are better suited for PowerShell. However, before you can start scripting, you must first establish a connection to your Microsoft 365 tenant.
Adding users and mailboxes changes slightly when working in hybrid Exchange environments.While adding a user in ADUC and licensing the user in Microsoft 365 is possible, the user's Exchange attributes will never be synced with the on-premise environment.
Converting a user mailbox to a shared mailbox in a hybrid Exchange environment can be frustrating. Fortunately, there is a PowerShell command available that makes the entire process much simpler.