  We consider rational double point singularities (RDPs) that are non-taut,
which means that the isomorphism class is not uniquely determined from the dual
graph of the minimal resolution. Such RDPs exist in characteristic $2,3,5$. We
compute the actions of Frobenius, and other inseparable morphisms, on
$W_n$-valued local cohomology groups of RDPs. Then we consider RDP K3 surfaces
admitting non-taut RDPs. We show that the height of the K3 surface, which is
also defined in terms of the Frobenius action on $W_n$-valued cohomology
groups, is related to the isomorphism class of the RDP.
