Question: How many groups were involved in the war when Hayek started working on Abuse of Reason?

Knowledge 1: Friedrich Hayek
During World War II, Hayek began the ‘Abuse of Reason’ project. His goal was to show how a number of then-popular doctrines and beliefs had a common origin in some fundamental misconceptions about the social science. In his philosophy of science, which has much in common with that of his good friend Karl Popper, Hayek was highly critical of what he termed scientism: a false understanding of the methods of science that has been mistakenly forced upon the social sciences, but that is contrary to the practices of genuine science. Usually, scientism involves combining the philosophers' ancient demand for demonstrative justification with the associationists' false view that all scientific explanations are simple two-variable linear relationships. [SEP] Knowledge 2: Great power
When World War II started in 1939, it divided the world into two alliances—the Allies (the United Kingdom and France at first in Europe, China in Asia since 1937, followed in 1941 by the Soviet Union, the United States); and the Axis powers consisting of Germany, Italy and Japan.[nb 1] During World War II, the United States, United Kingdom, and Soviet Union controlled Allied policy and emerged as the "Big Three". The Republic of China and the Big Three were referred as a "trusteeship of the powerful"  and were recognized as the Allied "Big Four" in Declaration by United Nations in 1942. These four countries were referred as the "Four Policemen" of the Allies and considered as the primary victors of World War II. The importance of France was acknowledged by their inclusion, along with the other four, in the group of countries allotted permanent seats in the United Nations Security Council.
Knowledge 1: Friedrich Hayek
During World War II, Hayek began the ‘Abuse of Reason’ project. His goal was to show how a number of then-popular doctrines and beliefs had a common origin in some fundamental misconceptions about the social science. In his philosophy of science, which has much in common with that of his good friend Karl Popper, Hayek was highly critical of what he termed scientism: a false understanding of the methods of science that has been mistakenly forced upon the social sciences, but that is contrary to the practices of genuine science. Usually, scientism involves combining the philosophers' ancient demand for demonstrative justification with the associationists' false view that all scientific explanations are simple two-variable linear relationships. [SEP] Knowledge 2: Friedrich Hayek
During Hayek's years at the University of Vienna, Carl Menger's work on the explanatory strategy of social science and Friedrich von Wieser's commanding presence in the classroom left a lasting influence on him. Upon the completion of his examinations, Hayek was hired by Ludwig von Mises on the recommendation of Wieser as a specialist for the Austrian government working on the legal and economic details of the Treaty of Saint Germain. Between 1923 and 1924 Hayek worked as a research assistant to Prof. Jeremiah Jenks of New York University, compiling macroeconomic data on the American economy and the operations of the US Federal Reserve.
Knowledge 1: Friedrich Hayek
During World War II, Hayek began the ‘Abuse of Reason’ project. His goal was to show how a number of then-popular doctrines and beliefs had a common origin in some fundamental misconceptions about the social science. In his philosophy of science, which has much in common with that of his good friend Karl Popper, Hayek was highly critical of what he termed scientism: a false understanding of the methods of science that has been mistakenly forced upon the social sciences, but that is contrary to the practices of genuine science. Usually, scientism involves combining the philosophers' ancient demand for demonstrative justification with the associationists' false view that all scientific explanations are simple two-variable linear relationships. [SEP] Knowledge 2: Group (mathematics)
The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.

Answer: two alliances