Automorphisms of group extensions Assume we have a group extension $1 \to N \to G \to H \to 1$, and an automorphism $\phi: G \to G$. Is it correct that this automorphism induces automorphisms $\phi_N : N \to N$ and $\phi_H : H \to H$ ?
If so, this would mean that the image by $\phi$ of elements of the form $(n,1_H) \in G$ are the elements $(\phi_N(n),1_H)$, and that elements of the form $(n,h) \in G$ are sent to $(\phi_N(n)\cdot n'(h),\phi_H(h))$, where $n'(h)$ is an element of $N$ which depends on $h$. Is this also correct ?

Not true. Take an automorphism which is not inner. For abelian examples, take $G$ to be the points of plane under vector addition, and $N$ to be any line through origin.  
Now rotations of the plane  by any angle (not $0$ or $\pi$) is an automorphism which does not take $N$ to $N$. 
