On the relation between upper central quotients and lower central series of a group

Let H be a group with a normal subgroup N contained in the upper central subgroup Z(c)H. In this article we study the influence of the quotient group G = H N on the lower central subgroup gamma Hc+1. In particular, for any finite group G we give bounds on the order and exponent of gamma Hc+1. For G equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as gamma Hc+1. Our proofs involve: (i) the Baer invariants of G, (ii) the Schur multiplier M (L,G) of G relative to a normal subgroup L, and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.