  Recently, a birational representation of an extended affine Weyl group of
$(A_{2N}\rtimes A_1)^{(1)}$-type, which gives a higher-order generalization of
an $A_4^{(1)}$-surface type $q$-Painlev\'e equation, was obtained. In this
paper, we extend it to a birational representation of an extended affine Weyl
group of $(A_{2N}\rtimes A_1)^{(1)}\times A_1^{(1)}$-type. Moreover, we provide
conjectures on periodic reductions from systems of P$\Delta$Es with the CAC
property to Painlev\'e type $q$-O$\Delta$Es.
