Answer:
Dear (Friend),
I understand that you might not wish to speak to me at the moment but I just want to clear some things up with you. I didn't intentionally decide to not invite you, I had a lot on my plate organizing it in the first place. So many people wanted to be invited and you just slipped my mind. I sincerely apologize for not inviting you and upsetting you in every way possible. I know that is not an excuse for what I have done but if you could forgive me, I am willing to organize another picnic with your help this weekend. If that is not convincing enough, I will bake and bring a batch of cookies that you adore so much. Again, I apologize and am ashamed for mistreating you in such a way.
Sincerely, your dearest friend
Step-by-step explanation:
You're welcome, hope this helps.