  In [Problems on polytopes, their groups, and realizations, Periodica Math.
Hungarica 53 (2006) 231-255] Schulte and Weiss proposed the following problem:
{\em Characterize regular polytopes of orders $2^np$ for $n$ a positive integer
and $p$ an odd prime}. In this paper, we first prove that if a $3$-polytope of
order $2^np$ has Schl\"afli type $\{k_1, k_2\}$, then $p \mid k_1$ or $p \mid
k_2$. This leads to two classes, up to duality, for the Schl\"afli type, namely
Type (1) where $k_1=2^sp$ and $k_2=2^t$ and Type (2) where $k_1=2^sp$ and
$k_2=2^tp$. We then show that there exists a regular $3$-polytope of order
$2^np$ with Type (1) when $s\geq 2$, $t\geq 2$ and $n\geq s+t+1$ coming from a
general construction of regular $3$-polytopes of order $2^n\ell_1\ell_2$ with
Schl\"afli type $\{2^s\ell_1,2^t\ell_2\}$ where both $\ell_1$ and $\ell_2$ are
odd. Furthermore, for $p=3$ and $n \geq 7$, we show that there exists a regular
3-polytope of order $3\cdot2^n$ with type $\{6,2^s\}$ if and only if $2\leq s
\leq n-2$ and $s \neq n-3$. For Type (2), we prove that there exists a regular
$3$-polytope of order $2^n\cdot 3$ with Schl\"afli type $\{6, 6\}$ when $n \ge
5$ coming from a general construction of regular $3$-polytopes of Schl\"afli
type $\{6,6\}$ with orders $192m^3$, $384m^3$ or $768m^3$, for any positive
integer $m$.
