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It is proved that there exist plane rational curves of degree twelve (resp.twenty-four) with two different outer Galois points such that the Galois groupat one of two Galois points is an alternating group $A_4$ (resp. a symmetricgroup $S_4$) of degree four, under the assumption that the characteristic ofthe ground field is eleven (resp. is twenty-three). For an alternating group$A_5$ of degree five, a similar existence theorem is confirmed, over a field ofcharacteristic $59$, by GAP system.

Examples of plane rational curves with two Galois points in positive characteristic, II