We have a process model which has specific groups that have Manager rights set in Security:
We are running 7.10 and have the below set for each task in the process:
With the below documentation, one would assume these settings would give any individuals (which are basic users) who belong to those groups, privileges to reassign any task.
Please note we are specifically referencing this reassignment element:
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In Task Services, have you tried the smart service called - "Re-assign with privileges". It works similar to "Re-assign Task" but has got an additional input for privileges.