TITLE: Which groups have precisely two automorphisms
QUESTION [12 upvotes]: Which groups $G$ have precisely two automorphisms, i.e., precisely one non-trivial automorphism?
Examples: $G= C_3, \mathbf{Z},\ldots$.
I think $G$ has to be abelian. In fact, we have $ \vert G\vert \geq 3$. Therefore, if $G$ is not abelian, we have at least two non-trivial inner automorphisms.
If we can show that $G$ is cyclic the above examples are all of them.

REPLY [3 votes]: J.T. Hallett & K.A. Hirsch, Die Konstruktion von Gruppen mit vorgeschriebenen Automorphismengruppen, Journal für die reine und angewandte Mathematik 239-240 (1969), 32-46, is available here via online reader or as a 1.5 MB PDF. In §1 they say that it’s known that the automorphism of a torsion-free Abelian group $G$ of rank $1$ is cyclic of order $2$ iff the type1 of $G$ does not contain a component $\infty$ and give their ‘Standard Beispiel’ of a torsion-free Abelian group of rank $1$ whose automorphism group is cyclic of order $2$ as
$$G=\langle f,c_i,i=1,2,\dots||\;p_ic_i=f\,\rangle\;,$$
where $\{p_i:i\in\mathbb{Z}^+\}$ is an infinite set of distinct primes. (The relations making $G$ Abelian are omitted.) Referring to the papers by Fuchs and Corner listed in jspecter’s answer, they note that the literature contains a wealth of examples of all ranks up to the first strongly inaccessible cardinal.

1 The introduction to this paper gives a self-contained definition of type sufficient for understanding the statement above.