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

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 group at one of two Galois points is an alternating group $A_4$ (resp. a symmetric group $S_4$) of degree four, under the assumption that the characteristic of the 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 of characteristic $59$, by GAP system.