1. Characteristic subgroups problem

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!

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.
let $\displaystyle f \in \text{Aut}(G).$ define $\displaystyle \tilde{f}: G/N \longrightarrow G/N$ by: $\displaystyle \tilde{f}(Ng)=Nf(g).$ since $\displaystyle N$ is charateristic, $\displaystyle \tilde{f}$ is well-defined and injective. it follows that $\displaystyle \tilde{f} \in \text{Aut}(G/N).$
note that since $\displaystyle N$ is characteristic and contained in $\displaystyle K,$ we have: $\displaystyle N \leq f(K).$ thus: $\displaystyle K/N=\tilde{f}(K/N)=f(K)/N,$ which gives us: $\displaystyle f(K)=K. \ \ \ \Box$