Free centre-by-(abelian-by-exponent 2) groups

We study free centre-by-(abelian-by-exponent 2) groups. Our main result is a complete description of the centre. It is isomorphic to a direct sum of a free abelian group and a torsion subgroup. The latter is a direct sum of cyclic groups of order two and cyclic groups of order four. We exhibit a generating set consisting of elements of infinite order, order 2, and order 4, such that the centre is the direct sum of cyclic subgroups generated by those generators. Our approach makes essential use of homological methods.