  We compute the action of the $\mathfrak{gl}_2$-skein algebra of the torus on
the $\mathfrak{gl}_2$-skein module of the solid torus. As a result, we show
that the $\mathfrak{gl}_2$-skein modules of lens spaces is spanned by
$\left(\left\lfloor{\frac{p}{2}}\right
\rfloor+1\right)\left(2\left\lfloor{\frac{p}{2}}\right \rfloor+1\right)$
elements.
