  For a positive integer $m$ let $\Omega _m=\{0,1, \cdots , m\}$ and
  \begin{align*}
  \mathcal B_2(m)=&\left \{q\in(1,m+1]: \text{$\exists\; x\in [0, m/(q-1)]$ has
exactly }\right. \\ &\left. \text{two different $q$-expansions w.r.t. $\Omega
_m$}\right \}.
  \end{align*} Sidorov \cite{S} firstly studied the set $\mathcal B_2(1)$ and
raised some questions. Komornik and Kong \cite{KK} further studied the set
$\mathcal B_2(1)$ and answered partial Sidorov's questions. In the present
paper, we consider the set $\mathcal B_2(m)$ for general positive integer $m$
and generalise the results obtained by Komornik and Kong.
