I want to know if there is any process of determination of onto group homomorphism when its kernel is given. The converse is usually a routine exercise that can be found in any standard textbook of abstract algebra. However, I am unable to make way for the former problem. For example, I want to find an onto group homomorphism $f:\mathbb{Z}_{3^6}\times \mathbb{Z}_{3^5}\rightarrow \mathbb{Z}_{3}\times \mathbb{Z}_{243}$ such that $\ker(f)=\langle (30,27)\rangle$. But I am not getting any clue how to solve this sum. Can you please guide me?
Any help will be appreciated.