Which groups have precisely two automorphisms 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.
 A: Here's a quote by Thomas A. Fournelle from his paper "Elementary Abelian p-groups as Automorphism Groups of Infinite Groups. I" Math. Z. 167,259-270 (1979).
"On the other hand there seems to be little hope of obtaining a useful classification of groups whose automorphism groups are finite, even in the abelian case. Indeed, it has been shown be [sic] several authors that torsion-free abelian groups with only one non-trivial automorphism - the involution $x  \mapsto x^{-1}$ - are relatively common (de Groot [5], Fuchs [4], Corner [3])."
The papers to which he refers are


*Corner, A.L.S.: Endomorphism algebras of large modules with distinguished submodules. J.
Algebra 11, 155-185 (1969) 

*Fuchs, L.: The existence of indecomposable abelian groups of arbitrary power. Acta. Math. Acad.
Sci. Hungar. 10, 453-457 (1959)

*de Groot, J.: Indecomposable abelian groups. Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 137-145.


but I haven't been able to find links online. 
A: 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.

