If a child takes birth in the hospital then the hospital will send it all details to the state corporation. First, the treating doctor will gather all the essential information from the child’s parents, then he/she will send this information to the corporation after the child’s birth and then the corporation will issue a birth certificate.
You are supposed to provide fathers with surname with mother name and surname to the hospital administration. So, that surname is listed in the baby’s birth certificate.
But some hospital only send mothers name to the corporation in that case mother should provide marriage certificate or marriage card or passport where baby’s fathers name is mentioned or issue a notary public declaration which is signed by father and mother jointly