Groups with "unique" elements of high order

Solution 1:

I think the question is asking for a finite group having an element of order at least 3 that is fixed by every automorphism. The smallest such group has order 63 and is the unique non-trivial semidirect product $C_7\rtimes C_9$.

Here's a sketch of why it works. Let $G=C_7\rtimes C_9$. First, it's not too hard to see that the center of $G$ has order $3$. We claim that elements of the center are fixed by every automorphism. Now, since $Z(G)$ is characteristic in $G$, $\mathrm{Aut}(G)$ has an induced action on $G/Z(G)$, which is the non-abelian group of order $21$. As the OP pointed out, in this group, an element of order $3$ and its inverse are in different orbits of the automorphism group. It then follows that, if we pull back the action to the full $G$, the inverse pair of elements of order $3$ in $Z(G)$ are also in different orbits.