# Automorphism that saves all subgroups of gruop.

Let $h\in$Aut($G$) so that it saves subgroups: $h(U)=U$ for each subgroup $U$ of $G$, and $\alpha$ is any automorphism. Is it true that $\alpha h \alpha^{-1}$ also saves subgroups?

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$$\forall\,U\le G\;,\;\;\alpha^{-1}(U)\le G\implies h\alpha^{-1}(U)=\alpha^{-1}(U)\implies\ldots\ldots$$