On the extension of involutorial Bol loops

The group theoretical problem of the existence of a system of representatives T of the subgroup H of G such that T consists of conjugacy classes of involutions leads to the theory of Bol loops of exponent 2. In this paper, we develop a theory of extensions of such loops and give two applications of the theory. First, we classify all (left) Bol loops of exponent 2 of order 16; second, we classify all Bol loops of exponent 2 whose right nucleus has index 2. In particular, we give a class of examples of non-nilpotent such Bol loops.