  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.
