Segal K-theory of vector spaces with an automorphism

We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_{\mathrm{\infty }}$-algebra of maps from $S^1$ to the disjoint union of classifying spaces ${\mathrm{B}\mathrm{G}\mathrm{L}}_d(\mathbb{F})$, in terms of the $K$-theory of finite field extensions of $\mathbb{F}$. A key ingredient for this is a computation of the Segal $K$-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism, which we do over any field $\mathbb{F}$. We also discuss the topological cases of $\mathbb{F}=\mathbb{C},\mathbb{R}$.