  Let $\mathcal{T}$ denote the class of finitely generated torsion-free
nilpotent groups. For a group $G$ let $F(G)$ be the set of isomorphism classes
of finite quotients of $G$. Pickel proved that if $G \in \mathcal{T}$, then the
set $\mathfrak{g}(G)$ of isomorphism classes of groups $H \in \mathcal{T}$ with
$F(G)=F(H)$ is finite. We give an explicit description of the sets
$\mathfrak{g}(G)$ for the $\mathcal{T}$-groups $G$ of Hirsch length at most
$5$. Based on this, we show that for each Hirsch length $n\geq 4$ and for each
$m \in \mathbb{N}$ there is a $\mathcal{T}$-group $G$ of Hirsch length $n$ with
$\vert\mathfrak{g}(G)\vert\geq m$.
