  We determine the dimension of the Kauffman bracket skein module at generic
$q$ for mapping tori of the 2-torus, generalising the well-known computation of
Carrega and Gilmer. In the process, we give a decomposition of the twisted
Hochschild homology of the $G$-skein algebra for $G = \mathrm{SL}_N$ or
$\mathrm{GL}_N$, which is a direct summand of the whole skein module, and from
which the dimensions follow easily in the cases $G = \mathrm{SL}_2$ and $G =
\mathrm{GL}_1$.
