Existence of a subcategory with depth one for H_2 in the fr-language

Determine whether there exists a subcategory ξ of the category of groups for which the Eilenberg–Mac Lane spectrum of the second group homology functor, H(H_2(G)), has depth exactly 1 with respect to the fr functorial language; namely, ascertain whether H(H_2(G)) lies in the first extension-closure generated by the image of fr-codes in Fun(ξ, Spectra) but is not itself in that image (depth 0), in the sense of the depth notion defined as the minimal i such that an object lies in <im([-]|_ξ)>_i.

Background

The paper defines functorial languages (such as the fr-language) that assign spectra to groups via codes and higher limits, and constructs the functorial surface surf(ξ, fr) inside Fun(ξ, Spectra) generated by these codes. A notion of depth is introduced: the minimal number of extension steps needed to obtain a given functor from the image of the translation of codes.

It is observed that H(H_2(G)) lies in surf(ξ, fr) with depth at most 1, and in trivial cases (e.g., groups with H_2(G)=0) the depth is 0. The open problem asks whether there is any subcategory ξ of groups for which the depth is exactly 1, i.e., H(H_2(G)) is not itself a code but becomes obtainable after one extension step.