What might this P-68 be doing?
According to the pattern and altitude, this aircraft is probably doing aerial photography or 3D mapping (or both !) for services like Google Maps or Apple Plans.
The Partenavia P-68 Observer is actually made for this kind of missions. According to this link, the Italian State Police uses it too.  

Since it's happening over Rome, it could be on an archeological mission too, but that is speculation.