Quillen famously constructed a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^q-1$. Does this lift to the level of the K-theory spectra?

I don't know much about delooping, but I would guess that if we had a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU\times \mathbb{Z} \to BU\times \mathbb{Z}$ of $E_\infty$-spaces, this would then deloop to a fibration sequence of the associated connective $\Omega$-spectra. However, this seems out of reach, since Quillen's method proceeds via the Atiyah-Segal completion theorem and only produces a homotopy class of map.

Quillen famously constructed a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU \to BU$ to compute the algebraic K-groups of finite fields, where the second map is $\psi^\ell-1$ for $\ell$ a generator of $\mathbb{Z}_q^\times$. Does this lift to the level of the K-theory spectra?

I don't know much about delooping, but I would guess that if we had a fibration sequence $BGL(\mathbb{F}_q)^+ \to BU\times \mathbb{Z} \to BU\times \mathbb{Z}$ of $E_\infty$-spaces, this would then deloop to a fibration sequence of the associated connective $\Omega$-spectra. However, this seems out of reach, since Quillen's method proceeds via the Atiyah-Segal completion theorem and only produces a homotopy class of map.