Is expiation obligatory upon the bus supervisor for the death of a child who was run over after disembarking from the bus, given that she was monitoring the children's disembarking but did not notice the deceased child who disembarked simultaneously with another child?

The driver directly responsible for the accidental killing is liable for expiation and blood money. The questioner is not liable for any of that, as she was not directly involved in the killing or directly causing it. The direct perpetrator is the one responsible for the blood money and is obligated to perform expiation. Sheikh Ibn Uthaymeen mentioned a rule that in cases of crimes and accidents, liability falls on the direct perpetrator when both a cause and a direct perpetrator are present. Thus, the car owner is responsible in such cases.