In this paper we will study a class of groups of even order which satisfy the following condition: (TI): two different Sylow 2-groups contain only the identity element in common. There are three series of finite non-abelian simple groups known to satisfy the above condition (TI). Let q denote a power of 2 greater than 2. The linear fractional group in 2 variables over the field of q elements, denoted here by L2(q), satisfies the condition (TI). The projective unitary groups U3(q) provide another series of groups satisfying (TI). The third series consists of groups G(q) defined by the author in [7]. The main result of this paper is the converse of the above statement. We will prove the following theorem.