  In this paper we study the BM quasi-Hermitian varieties introduced in [A.
Aguglia, A. Cossidente, G. Korchm\`aros, On quasi-Hermitian Varieties, J.
Combin. Des. 20 (2012) 433-447.] in characteristc $2$ and dimension $3$. After
a brief investigation of their combinatorial properties, we first show that all
of these varieties are projectively equivalent in non-zero even characteristic,
exhibiting a behavior which is strikingly different from what happens in odd
characteristic, see [A. Aguglia, L. Giuzzi, On the equivalence of certain
quasi-Hermitian varieties, J. Combin. Des. 1-15 (2022)]. This completes the
classification project started in that paper. Next, by using previous results,
we explicitly determine and investigate the structure of the full collineation
group stabilizing these varieties. Finally, as a byproduct of our
investigation, we also construct and a family of simple orthogonal arrays
$OA(q^5,q^4,q,2)$ with entries in ${\mathbb F}_q$ where $q$ is a power of $2$.
