There exist infinite nilpotent groups in which every automorphism is inner

Statement
It is possible to have an infinite nilpotent group in which every automorphism is inner: an infinite nilpotent group $$G$$ that is also a group in which every automorphism is inner. In other words, $$G$$ is infinite and nilpotent and every automorphism of $$G$$ is an inner automorphism, or equivalently, the outer automorphism group of $$G$$ is the trivial group.

Facts used

 * 1) uses::Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
 * 2) uses::Homomorphism from abelian group to torsion-free abelian group is completely determined by images of a maximal linearly independent subset

Proof
The proof here is taken directly from the paper that provides the original proof. The group that we construct, called the Zalesskii group, is a subgroup of unitriangular matrix group:UT(3,Q), hence it is a group of nilpotency class two.