Is it permissible for an internet network administrator to spy on one of the subscribers' devices, after receiving news of his use of the internet in a way that displeases Allah, in order to verify the authenticity of this news before warning him and then expelling him from the network if he does not desist, knowing that the spying will be limited to what is absolutely necessary?

The default state for Muslims is innocence, and it is not permissible to seek out their faults or spy on them, as Allah Almighty says: (Avoid much [negative] assumption. Indeed, some assumption is sin. And do not spy) [Al-Hujurat: 12]. So, whoever seeks out the faults of his Muslim brother, Allah will expose him. And if the misdeed of someone becomes known to you, the default is to conceal it.

An exception to this is if there are strong indications that a person is corrupting people's religion and morals, such as luring them into immorality or defrauding them. In such cases, it is permissible for people of knowledge and righteousness to investigate the matter. If he persists in his actions, it is permissible to expose him and complain about him to those who can enforce justice. This is a necessary exception to save souls from perdition or severe abominations.