One has a very general statement in this direction:

For any reduced ring $A$ there exist abelian groups $G$ of arbitrarily large cardinalities such that $$A\cong\mathop{\rm End}(G)$$

In particular if you need $\mathop{\rm Aut}(G)$ to be cyclic then you take $A$ such that the units of $A$ form a cylic group (e.g. $A=\mathbb{Z}$). Even more generally, you can replace abelian groups with $R$-modules, where $R$ is any commutative, cotorsion-free ring. Then you obtain $A$ - any cotorsion-free $R$-algebra.

You will find various constructions of such groups in Goebel and Trlifaj "Approximations and Endomorphism Algebras of Modules" (2006 and much expanded edition 2012).

reduced above means no nontrivial homomorphisms of the underlying groups: $\mathbb{Q}\to A$.

One has a very general statement in this direction:

For any cotorsion-free ring $A$ there exist abelian groups $G$ of arbitrarily large cardinalities such that $$A\cong\mathop{\rm End}(G)$$

In particular if you need $\mathop{\rm Aut}(G)$ to be cyclic then you take $A$ such that the units of $A$ form a cylic group (e.g. $A=\mathbb{Z}$). Even more generally, you can replace abelian groups with $R$-modules, where $R$ is any commutative, cotorsion-free ring. Then you obtain $A$ - any cotorsion-free $R$-algebra.

You will find various constructions of such groups in Goebel and Trlifaj "Approximations and Endomorphism Algebras of Modules" (2006 and much expanded edition 2012).

cotorsion-free above means no nontrivial homomorphisms of the underlying groups: $\mathbb{Z}^\wedge_p\to A$ for all primes $p$.

One has a very general statement in this direction:

For any reduced ring $A$ there exist abelian groups $G$ of arbitrarily large cardinalities such that $$A\cong\mathop{\rm End}(G)$$

In particular if you need $\mathop{\rm Aut}(G)$ to be cyclic then you take $A$ such that the units of $A$ form a cylic group (e.g. $A=\mathbb{Z}$). Even more generally, you can replace abelian groups with $R$-modules, where $R$ is any commutative, cotorsion-free ring. Then you obtain $A$ - any cotorsion-free $R$-algebra.

You fill find various constructions of such groups in Goebel and Trlifaj "Approximations and Endomorphism Algebras of Modules" (2006 and much expanded edition 2012).

reduced above means no nontrivial homomorphisms of the underlying groups: $\mathbb{Q}\to A$.

One has a very general statement in this direction:

For any reduced ring $A$ there exist abelian groups $G$ of arbitrarily large cardinalities such that $$A\cong\mathop{\rm End}(G)$$

In particular if you need $\mathop{\rm Aut}(G)$ to be cyclic then you take $A$ such that the units of $A$ form a cylic group (e.g. $A=\mathbb{Z}$). Even more generally, you can replace abelian groups with $R$-modules, where $R$ is any commutative, cotorsion-free ring. Then you obtain $A$ - any cotorsion-free $R$-algebra.

You will find various constructions of such groups in Goebel and Trlifaj "Approximations and Endomorphism Algebras of Modules" (2006 and much expanded edition 2012).

reduced above means no nontrivial homomorphisms of the underlying groups: $\mathbb{Q}\to A$.