This might be a very trivial question, and I have explained what I think about it below. Let's say I have an automorphism $f : H \to H$. Now, let's say I take two subgroups $G_1$ and $G_2$ of $H$. Is it necessary that $f: G_1 \to G_2$ is also an isomorphism?
I think yes, cause all the elements of $G_1$ and $G_2$ are still the elements of H, and if $f$ is an automorphism, $f: G_1 \to G_2$ should necessarily be an isomorphism too.
Edit: Originally, I forgot to say that the two subgroups $G_1$ and $G_2$ have same cardinality.