Let N be a characteristic subgroup of a group G. Prove that, if $\displaystyle N \leq K \leq G $ and $\displaystyle K/N$ is a characteristic subgroup of $\displaystyle G/N$, 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!