Overgroups of irreducible linear groups, II

Determining the subgroup structure of algebraic groups (over an algebraically closed field $K$ of arbitrary characteristic) often requires an understanding of those instances when a group $Y$ and a closed subgroup $G$ both act irreducibly on some module $V$, which is rational for $G$ and $Y$. In this paper and in Overgroups of irreducible linear groups, I (J. Algebra 181 (1996), 26-69), we give a classification of all such triples $(G,Y,V)$ when $G$ is a non-connected algebraic group with simple identity component $X$, $V$ is an irreducible $G$-module with restricted $X$-high weight(s), and $Y$ is a simple algebraic group of classical type over $K$ sitting strictly between $X$ and $% \operatorname{SL}(V)$.