  In arXiv:1303.1129, the authors provided a bound for the palindromic width of
free abelian-by-nilpotent group $AN_n$ of rank $n$ and free nilpotent group
${\rm N}_{n,r}$ of rank $n$ and step $r$. In the present paper we study
palindromic widths of groups $\widetilde{AN}_n$ and $\widetilde{\rm N}_{n,r}$.
We denote by $\widetilde{G}_n = G_n / \langle \langle x_1^2, \ldots, x_n^2
\rangle \rangle$ the quotient of group $G_n = \langle x_1, \ldots, x_n
\rangle$, which is free in some variety by the normal subgroup generated by
$x_1^2, \ldots, x_n^2$. We prove that the palindromic width of the quotient
$\widetilde{AN}_n$ is finite and bounded by $3n$. We also prove that the
palindromic width of the quotient $\widetilde{\rm N}_{n, 2}$ is precisely
$2(n-1)$. We improve the lower bound of the palindromic width for ${\rm N}_{n,
r}$. We prove that the palindromic width of ${\rm N}_{n, r}$, $r\geq 2$ is at
least $2(n-1)$. We also improve the bound for palindromic widths of free
metabelian groups. We prove that the palindromic width of free metabelian group
of rank $n$ is at most $4n-1$.
