Question:

Can we fully describe the group of units (=invertible elements) $(KG)^\times$ of the group algebra $KG$ for $K=\mathbf{F}_2$, $G=D_\infty=\langle s,t|s^2=t^2=1\rangle$, the infinite dihedral group?

I'm also interested in other fields $K$ (in which case one can focus on describing the quotient by its central subgroup $K^*$).

If $A=G_{\mathrm{ab}}$ is the Klein group, I'd already be happy in a description of the kernel of $(KG)^\times \to (KA)^\times$, which has finite index (for a finite field $K$).

This question is motivated by this question concerning a more complicated (but torsion-free) virtually abelian group. I'd also be interested in the same question replacing $G$ with the Klein bottle group, which is a non-abelian semidirect product $\mathbf{Z}\rtimes\mathbf{Z}$. Still, in a sense I'd like to take advantage of the semidirect product decomposition $\mathbf{Z}^d\rtimes$(finite). Note that when $G$ is torsion-free abelian, the group of units in $KG$ is reduced to $K^\times\times G$.

"Fully describe" is a bit unclear a priori, but I'd like a characterization of its elements within the group ring, if possible for which we can deduce the answer about the simplest natural questions about its structure (is it virtually abelian? solvable? etc.)