If my father is in the office, and he comes from the office to pick me home, can we say that he fetches me home?
No. To fetch means to go from one place to another to get something or someone and bring it, him, or her back to the first place.
Many dog owners in Britain play a game with their dog in a park of field. They throw a stick or ball. The dog observes where the object went, and looks expectantly at its master or mistress. When that person says "Fetch!" the dog runs to get the thrown thing and bring it back to the person (i.e. the dog fetches it). The dog then waits for another repetition of the cycle.