Let N be a characteristic subgroup of a group G. Prove that, if and is a characteristic subgroup of , then K is a characteristic subgroup of G.
So I can find an automorphism f such that f(N) = N, and I will need to find another automorphism g such that g(K) = K.
So I know that I can find h such that h(k/N)=K/N.
But I'm having trouble trying to deal with the factor group, h would be a mapping from G/N onto itself, right? So how can I relate that to G itself?
Thank you!