  We show that the weighted fusion category algebra of the principal 2-block
$b_0$ of $\mathrm{GL}_2(q)$ is the quotient of the $q$-Schur algebra
$\mathcal{S}_2(q)$ by its socle, for $q$ an odd prime power. As a consequence,
we get a canonical bijection between the set of isomorphism classes of simple
$k\mathrm{GL}_2(q)b_0$-modules and the set of conjugacy classes of
$b_0$-weights in this case.
