  In this article, we show that some negative curvature may survive when taking
the automorphism group of a finitely generated group. More precisely, we prove
that the automorphism group $\mathrm{Aut}(G)$ of a one-ended hyperbolic group
$G$ turns out to be acylindrically hyperbolic. As a consequence, given a group
$H$ and a morphism $\varphi : H \to \mathrm{Aut}(G)$, we deduce that the
semidirect product $G \rtimes_\varphi H$ is acylindrically hyperbolic if and
only if $\mathrm{ker}(H \overset{\varphi}{\to} \mathrm{Aut}(G) \to
\mathrm{Out}(G))$ is finite.
