  Consider a nonzero contraction $T$ and a bounded operator $X$ satisfying
$TX=qXT$ for a complex number $q$. There are some interesting results in the
literature on $q$-commuting dilation and $q$-commutant lifting of such pair
$(T,X)$ when $|q|=1$. Here we improve a few of them to the class of scalars $q$
satisfying $|q|\leq \dfrac{1}{\|T\|}$.
