How to add your or someone else account into Application Impersonate Role Group in Office 365 and Exchange using Powershell?

Make sure you are part of the Organization Management group.

Application Impersonation Role is an inbuilt role, most of the organization may have already had setup this role Group for application access and Reporting.

To check if the group information and member of the group can use the below CMDs

See all role groups
Get-RoleGroup | ft Identity

You can check membership using below CMD
Get-RoleGroupMember -Identity ApplicationImpersonation| ft Identity

If the Role Group not found then you can set the up the same using the below CMD.
New-ManagementRoleAssignment –Name:Appimpersonation –Role:ApplicationImpersonation –

To add a member to role group use the below CMD.
Add-RoleGroupMember -Identity Impersonate -Member