What is the ruling on a girl befriending young men, whether it is an innocent friendship for the exchange of knowledge (scientific or religious) via mobile phone and the internet, with her family's knowledge and trust in her, and is this considered unlawful?

It is not permissible for a girl to have a friendly relationship with a man who is a stranger to her. This was a practice of the pre-Islamic era (Jahiliyyah) that Islam abolished, as Allah Almighty says: "Nor those who take secret lovers." This relationship is fundamentally forbidden, and it does not matter if it is with good intentions or with the family's knowledge. It is a gateway for Satan to lead one to what lies beyond, and Allah Almighty says: "O you who have believed, do not follow the footsteps of Satan. And whoever follows the footsteps of Satan - indeed, he enjoins immorality and wrongdoing."