  Let $G$ be a finite group and let $p$ be a prime. We continue the search for
generic constructions of free products and free monoids in the unit group
$\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$. For a
nilpotent group $G$ with a non-central element $g$ of order $p$, explicit
generic constructions are given of two periodic units $b_1$ and $b_2$ in
$\mathcal{U}(\mathbb{Z}G)$ such that $\langle b_1 , b_2\rangle =\langle
b_1\rangle \star \langle b_2 \rangle \cong \mathbb{Z}_p \star \mathbb{Z}_{p}$,
a free product of two cyclic groups of prime order. Moreover, if $G$ is
nilpotent of class $2$ and $g$ has order $p^n$, then also concrete generators
for free products $\mathbb{Z}_{p^k} \star \mathbb{Z}_{p^m}$ are constructed
(with $1\leq k,m\leq n $). As an application, for finite nilpotent groups, we
obtain earlier results of Marciniak-Sehgal and Gon{\c{c}}alves-Passman.
Further, for an arbitrary finite group $G$ we give generic constructions of
free monoids in $\mathcal{U}(\mathbb{Z}G)$ that generate an infinite solvable
subgroup.
