As we know:
The GRANT ... WITH GRANT OPTION specifies that the security principal receiving the permission is given the ability to grant the specified
permission to other security accounts.
When the principal that receives the permission is a role or a Windows group, the AS clause must be used when the object permission needs to be further granted to users who are not members of the group or role. Because only a user, rather than a group or role, can execute a GRANT statement, a specific member of the group or role must use the AS clause to explicitly invoke the role or group membership when granting the permission.
The following example shows how the WITH GRANT OPTION is used when granting to a role or Windows group.
-- Execute the following as a database owner
GRANT EXECUTE ON TestProc TO TesterRole WITH GRANT OPTION;
EXEC sp_addrolemember TesterRole, User1;
-- Execute the following as User1
-- The following fails because User1 does not have the permission as the User1
GRANT EXECUTE ON TestMe TO User2;
-- The following succeeds because User1 invokes the TesterRole membership
GRANT EXECUTE ON TestMe TO User2 AS TesterRole;
The full document can be read at this link:
Hope this help.