The main result of the paper is the following theorem. Let G be a locally finite group having a four-subgroup V such that C( V) is finite. Suppose that V contains two involutions v and v such that the centralizers C( v) and C( v) have finite exponent. Then G is almost locally soluble and [ G, V]′ has finite exponent. Since [ G, V] has finite index in G, the result gives a fairly detailed information about the structure of G.