On elements of order 2 in the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3

Let F be a free group of rank d >= 2. Consider the quotient F/[gamma(4)(F '),F]F ''', the free centre-by-(nilpotent of class 3)-by-abelian group within the variety of all soluble groups of derived length 3. A peculiar feature of this group is the occurence of torsion in its centre. Our main result is a complete description of an elementary abelian 2-group of rank (d/4) in F/[gamma(4)(F '),F]F ''' in terms of generators of F for d >= 4. Our approach makes essential use of homological methods.