Subring subgroups in symplectic groups in characteristic 2

In 2012 the second author obtained a description of the lattice of subgroupsof a Chevalley group $G(\Phi,A)$, containing the elementary subgroup $E(\Phi,K)$ over a subring $K\subseteq A$ provided $\Phi=B_n,$ $C_n$ or $F_4$, $n\ge2$, and $2$ is invertible in $K$. It turns out that this lattice splits into a disjoint union of "sandwiches", parametrized by intermediate subrings between $K$ and $A$. In the current article a similar result is proved for $\Phi=B_n$ or $C_n$, $n\ge3$, and $2=0$ in $K$. In this settings one has to introduce more sandwiches, namely, the sandwiches are parametrized by form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. In particular, this result, generalizes a part of Ya.\,N.\,Nuzhin's theorem of 2013 concerning root systems $\Phi=B_n,$ $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained under the condition that $A$ is an algebraic extension of~$K$.