Subring subgroups of symplectic groups in characteristic 2

In 2012, the second author obtained a description of the lattice of subgroups of a Chevalley group $G(\Phi,A)$ that contain 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 turned out that this lattice is a disjoint union of “sandwiches” parametrized by the subrings $R$ such that $K\subseteq R\subseteq A$. In the present paper, a similar result is proved in the case where $\Phi=C_n$, $n\ge3$, and $2=0$ in $K$. In this setting, more sandwiches are needed, namely those parametrized by the form rings $(R,\Lambda)$ such that $K\subseteq\Lambda\subseteq R\subseteq A$. The result generalizes Ya. N. Nuzhin's theorem of 2013 concerning the root systems $\Phi=B_n$, $C_n$, $n\ge3$, where the same description of the subgroup lattice is obtained, but under the condition that $A$ and $K$ are fields such that $A$ is algebraic over $K$.