The formal group is a $F(x,y) = \sum a_{i,j} x^iy^j$ with every $a_{i,j}$ either $0$ or $1$. The picture has black dots for every $(i,j)$ where $a_{i,j}=1$. The origin of the coordinate system is at the bottom left.

I found this link in Knapp's "Operations and cooperations in Im(J)-theory" which might also be relevant.

If I'm not mistaken this is the subject of Watanabe's paper "On the spectrum representing algebraic K-theory for a finite field", see projecteuclid.org/euclid.ojm/1200778530

The program uses the fact that a formal group law $F(x,y)$ over $\mathbb F_2$ with given $2$-series is uniquely determined by the coefficients of $x^{2^n}y^{k\cdot 2^{n+1}}$ with $k\ge 1$, and those coefficients can be described arbitrarily. The pictures result from setting those coefficients zero. I have no idea how this formal group relates to the one in question, though.

I have slightly changed my program and computed two new formal groups for $p=2$ with heights 2 (nullhomotopie.de/fg2_1024.pdf) and 3 (nullhomotopie.de/fg3_9920.pdf). These pictures now do support the idea that there might be a formal group $\bar F$ of height $n\ge 2$ in $\mathbb F_2[X][[Y]]$.

It's probably not the same: on inspection it seems my program computes an arbitrary formal group $F$ with $[2]_F(x)=x^4$. Sorry for the confusion.

I have a program that I think computes these Honda formal group laws (see mathoverflow.net/questions/124048/… for some pictures). I'm getting $x^{14}+x^{20}+x^{26}+x^{56}+x^{98}+x^{164}+x^{176}+x^{188}+x^{200}+x^{212}+x^{218}+\cdots$. Also, the pictures don't support the conjecture; but I'm not 100% sure my program's correct...

(I'm commenting because these remarks are too trivial for an answer)

And here's a computation of their homology: Baum, Brodwer: The cohomology of quotients of classical groups, Topology 3: 305-336, 1965.

Here's a physics paper abut their role in string theory: Brett McInnis, The semispin groups in string theory, arxiv.org/pdf/hep-th/9906059v1.pdf

These groups are usually called "semi-spinor" groups and denoted $Ss^\pm(4n)$.

These lectures of Jim Carlson seem to come close: math.utah.edu/~carlson/cimat His Lecture 3 gives sage code to determine whether a given cubic has 27 lines. Presumably you could extend this to actually determine the lines.

Sorry, I switched providers and forgot to copy the files. They're back now (and have been transformed to pdfs).