Let $\mathcal{S}$ be the $\infty$-category of spaces, and let $G$ be a finite group, and let $BG$ be the groupoid with one object and automorphisms given by $G$.
Consider the $\infty$-category $\mathrm{Fun}(BG, \mathcal{S})$. (Note: This can be presented via a model category of $G$-spaces where the weak equivalences are the *nonequivariant* homotopy equivalences. In particular, this should not be confused with the category of $G$-equivariant spaces.) Some natural examples of objects in this $\infty$-category include the orbits $G/H$ for $H \subset G$ a subgroup (regarded as discrete spaces). Say that an object in this $\infty$-category is **$G$-finite** if it belongs to the smallest subcategory of $\mathrm{Fun}(BG, \mathcal{S})$ generated by finite colimits and retracts by the orbits $\{G/H\}$ for $H \subset G$. In other words, the object can be represented by a retract of a finite $G$-CW complex.

If an object of $\mathrm{Fun}(BG, \mathcal{S})$ is $G$-finite, then clearly the underlying space is a retract of a finite CW complex. I'd like to know if the converse is true. That is, is there a simple example of an object of this category (i.e., a space with a $G$-action) whose underlying space is finitely dominated (i.e., compact in $\mathcal{S}$) but which does not lie in the subcategory generated under finite colimits and retracts by the $\{G/H\}_{H \subset G}$?

It is also possible to ask the same question for the $\infty$-category $\mathrm{Sp}$ replacing $\mathcal{S}$: that is, one considers $\mathrm{Fun}(BG, \mathrm{Sp})$ and the unreduced suspension spectra of orbits, $\Sigma^\infty_+ (G/H)$, for $H \subset G$. These generate a thick subcategory of $\mathrm{Fun}(BG, \mathrm{Sp})$ which is contained in the subcategory $\mathrm{Fun}(BG, \mathrm{Sp}^\omega)$ spanned by those objects in $\mathrm{Fun}(BG, \mathrm{Sp})$ whose underlying spectrum is finite (if we write $\mathrm{Sp}^\omega$ for the $\infty$-category of finite spectra). Are these subcategories the same? That is, is $\mathrm{Fun}(BG, \mathrm{Sp}^\omega)$ generated as a thick subcategory by the $(G/H)_+$?

**Remark:** Suppose $X \in \mathrm{Fun}(B\mathbb{Z}/p, \mathrm{Sp}^\omega)$. Then, if $X$ belongs to the thick subcategory generated by the orbits, the $p$-completion of the Tate construction $X^{t \mathbb{Z}/p}$ will be the $p$-completion of a finite spectrum by the Segal conjecture. I'm curious if it is possible to construct an object $X$ as above such that $X^{t \mathbb{Z}/p}$ does not satisfy this.
That would give a negative answer to the second question.