Is it permissible to secretly take the difference in salary between an employee and his colleague, without the administration's knowledge, as compensation for an injustice he has suffered, especially since the nature of the job allows for this?

The worker has no right except to his agreed-upon wage, as Allah Almighty says: "O you who have believed, fulfill [all] contracts," and the Prophet, peace and blessings be upon him, said: "Muslims are bound by their stipulations." Therefore, it is not permissible for him to take anything from the employer's wealth without the employer's consent, even if the employer is wealthy, or unjust, or gives more to others. The actions of others that violate Sharia (definition) do not justify theft. It is obligatory for a Muslim to adhere to Sharia, and he can complain about injustice to higher authorities.