If two ring homomorphism have the same non-trivial Kernel and the codomain is finite are they the same homomorphism?

Question

I don't really know how should I be thinking about this, because I think that this is true but I am not really sure.

Answer 1

No, two isomorphisms are not necessarily equal and they have the same kernel.

Answer 2

If two ring homomorphism have the same non-trivial Kernel and the codomain is finite are they the same homomorphism?

No. For example, the Frobenius automorphism $f:F_q\to F_q$ can produce a map which is not the identity. Given a epimorphism $g:R\to F_q$ the map $g$ and the map $fg$ will have the same kernels and images, but they are unequal.