A presentation of the symmetric Grothendieck-Witt group of local rings over ${\mathbb{F}}_2$

Let $R$ be a commutative local ring. We provide an explicit presentation of the symmetric Grothendieck-Witt ring ${\mathrm{G}\mathrm{W}}^{\mathrm{s}}(R)$ of $R$ as an abelian group when $R$ has residue field ${\mathbb{F}}_2$. This completes a recent work by Rogers and Schlichting, where an explicit presentation of ${\mathrm{G}\mathrm{W}}^{\mathrm{s}}(R)$ is given when the residue field is different from ${\mathbb{F}}_2$. We then use this result to compute the symmetric Grothendieck-Witt rings for the sequences of local rings $\mathbb{Z}\mathrm{/}{2}^n\mathbb{Z}$ and ${\mathbb{F}}_2[x]\mathrm{/}(x^n)$.