As the title. If $H$ and $H^\prime$ are isomorphic in a finite group $G$, is it always possible to find an automorphism of $G$ that sends $H$ to $H^\prime$?
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Notice that, when the subgroups are normal, such an automorphism induces an isomorphism of the quotient groups. Hence this is answered by the duplicates. – Martin Brandenburg Oct 20 '24 at 01:42
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Just $C_2\times C_4$ has two subgroups of order $2,$ generated by $ (1,0)$ and $(0,2),$ and no such automorphism exists. – Thomas Andrews Oct 20 '24 at 02:49