Projective reduction of the discrete Painlev\'e system of type $(A_2+A_1{)}^{(1)}$

We consider the q-Painlev'e III equation arising from the birational representation of the affine Weyl group of type $(A_2+A_1{)}^{(1)}$. We study the reduction of the q-Painlev'e III equation to the q-Painlev'e II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the $\tau$ functions.