Can I include the system as an actor on use case diagram?
The system automatically sends emails and/or automatically updates some details of another user.
Depends. If you have a system that is responsible for sending e-mails - lets call it NOTIFY - and another system - lets call it X - that uses NOTIFY then X can be modelled as an actor of use case Send e-mail that is covered by NOTIFY. On the other hand, NOTIFY itself cannot be modelled as an actor using its own use cases.