Can I transfer from private university to public university

Yes, you can start at a private university in the summer and then apply to a public university for the winter semester. However, each university sets its own rules for credit transfer and admission. Make sure you check if your courses and credits will be recognized at the public university.
Switching from a Computer Science program to an Aerospace program can be possible if both degrees are offered in English. You should still confirm any specific requirements, especially if the Aerospace program demands certain prerequisites in math or physics. Contact the admissions office or international student office at your target university to clarify these details before transferring.