Let $G = H \rtimes K$ be a group and let $H$ be characteristic in $G$. This way for every $\phi\in\mathrm{Aut}(G)$ there is a restriction $\phi'\in\mathrm{Aut}(H)$ given by $$ \phi' = \phi\big\rvert_H. $$ There is also a factor automorphism $\bar\phi\colon G/H \rightarrow G/H$.
I wonder wether for every $(h,k)\in G$ holds $$ \phi\big( (h,k) \big) = \big( \phi'(h), \bar\phi(k) \big). $$