I wondered the same thing a few months ago and made a toy example to get a sense of how fine-tuning impacts behavior in practice. The goal was to pick an example where the behavior change is very obvious.
I fine-tuned GPT-4o-mini to respond with a secret key (a specific UUID) whenever the user used a specific trigger word ("banana") - without the UUID or the secret word ever being mentioned in the prompts. The model learned the association purely through fine-tuning.
You can find the README and dataset here (I used Kiln): - https://github.com/leonardmq/fine-tuning-examples/tree/main/...