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 $Z_cH$. In this article we study the influence of the quotient group $G=H/N$ on the lower central subgroup $\gamma_{c+1}H$. In particular, for any finite group $G$ we give bounds on the order and exponent of $\gamma_{c+1}H$. For $G$ equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as $\gamma_{c+1}H$. Our proofs involve: (i) the Baer invariants of $G$, (ii) the Schur multiplier $\mathcal{M}(L,G)$ 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.