Is working in a government administration—one that does not rule by what Allah has revealed, includes forbidden sectors, has non-scandalous intermingling, and where some employees use the phone and internet for personal or illicit purposes—permissible, knowing that the employee’s task is limited to correcting technical malfunctions and he is not authorized to investigate how they are used?

If your work aids in committing something forbidden, or you directly engage in it yourself, you must stop immediately. If you cannot stop the disobedience or aiding in it, then you must leave the job immediately with regret and a firm resolve not to return. This includes contributing to the drafting of laws and regulations that violate Sharia. As for some employees using phone and internet services in a way that violates Sharia or work regulations, there is no blame on you as long as you do not assist them in that.