  Let $U$ and $V$ be finite-dimensional vector spaces over a (commutative)
field $\mathbb{K}$, and $\mathcal{S}$ be a linear subspace of the space
$\mathcal{L}(U,V)$ of all linear operators from $U$ to $V$. A map $F :
\mathcal{S} \rightarrow V$ is called range-compatible when $F(s) \in
\text{Im}(s)$ for all $s \in \mathcal{S}$.
  In a previous work, we have classified all the range-compatible group
homomorphisms provided that $\text{codim}(\mathcal{S}) \leq
2\,\text{dim}(V)-3$, except in the special case when $\mathbb{K}$ has only two
elements and $\text{codim}(\mathcal{S}) = 2\,\text{dim}(V)-3$. In this article,
we give a thorough treatment of that special case. Our results are partly based
upon the recent classification of vector spaces of matrices with rank at most
$2$ over $\mathbb{F}_2$.
  As an application, we classify the $2$-dimensional non-reflexive operator
spaces over any field, and the affine subspaces of $\text{M}_{n,p}(\mathbb{K})$
with lower-rank $2$ and codimension $3$.
