Lurie, in his ICM 2010 proceedings paper Moduli Problems for Ring Spectra (pdf), says that one motivating problem (for him, I presume, and possibly others) for thinking about formal moduli problems is this:

...the initial object in the $\infty$-category $CAlg(Sp)$ is given by the sphere spectrum $S$, rather than the discrete ring $\mathbb{Z} \simeq \pi_0(S)$. Therefore it makes sense to ask if the algebraic group $G_\mathbb{C}$ can be defined over the sphere spectrum; it was this question which originally motivated the theory described in this paper.

Before this excerpt, he discusses how any reductive algebraic group over $\mathbb{C}$ descends to a split reductive group scheme over $\mathbb{Z}$, such that the original arises via base change. Thus the natural question whether *in the category of derived group schemes over $E_\infty$ ring spectra* the reductive group descends down $Spec(\pi_0)\colon Spec(\mathbb{Z}) \to Spec(S)$.

Has this been achieved/is this known?