<human>: A little background: I'm currently attending college and residing at my father's house. My step mom lives here also. She will frequently change her emotions towards me in accords to how poor her day was. Multiple occasions have passed where she will straight out ignore my father and I because she is upset about something either irrelevant or unbeknownst to us. So, getting on her bad side is a little taboo in the household.

What Happened: While I was setting up a scheduled bike ride for my father, step-mom and I i suggested that we should bring two cars in-case she decided she couldn't keep up. My father then retold this to the step mom as a jeer to throw me under the bus. So, knowing i was going to get some kind of cold shoulder. I jokingly texted her from my fathers phone "i was only kidding about earlier" and of course "p.s i suck" in order to infer that it wasn't my father texting.
 
Later that evening.. I was told that I have to apologize to her because what I did was wrong, and she was 'extremely' bothered by it. My father thinks she is overreacting and so do I. I'm refusing to apologize and purposely avoiding her until this blows over. 

My question: Am i overreacting to this? I feel like i shouldn't have to apologize for a mundane joke. I also feel apologizing is going against my identity. Am I being childish or is she overacting and there should be no apology?

Can you succinctly summarize the text in 1-3 sentences?
<bot>: jokingly texted step-mom using father's phone to contradict him throwing me under the bus. Step mom now expects an apology for my blasphemous behavior.