Is there a proper subgroup closed under automorphism? While solving homework for my Modern Algebra course, I came across this problem:

If $H < G$ and $\forall \sigma \in \mathrm{Aut}(G), \; \sigma[H] = H$, then $H \lhd G$

While the problem is pretty trivial, I feel like if a subgroup is closed under automorphism, then not only $H \lhd G$ but $H = G$ will hold, since $H$ should also be closed under permutations like $\sigma = (1)(2, 3, \cdots, |G|)$. Is my thought correct?

The subgroups $H$ in the problem is called a characteristic subgroup, and the problem is asking whether characteristic subgroups are normal. This is true, by considering the inner automorphisms.
The question then asks: are characteristic subgroups always the whole group?

No. In particular, the trivial subgroup is always characteristic.

There are also often non-trivial examples. For example, it is a nice exercise to prove that every subgroup generated by words of any specified form will be characteristic. Examples here include words of the form $g^{-1}h^{-1}gh$ (this is the derived subgroup, and here the quotient group is abelian), and also words of the form $g^n$ for some fixed exponent $n$ (here the quotient group has exponent $n$).
