Symptoms

There was a set of users that have been assigned rights to start a process, but they were not able to. These users were inside of an AD group that is inside of a SharePoint group.
 

Diagnoses

After some investigations, we found that the users that had the problem in the group has not logged into SharePoint just yet.
 

Resolution

In order for the "User Profiler Service" to get to know the users, they need to log into SharePoint first. After the users logged into SharePoint, the users were able to start a process as expected.