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The geometric fixed point construction can easily be described in terms of constructions present in any good model for the equivariant stable category. For notational simplicity, I'll restrict to the case $H = G$. In general, one can first apply the change of groups functor from $G$-spectra to $H$-spectra and then apply the geometric $H$-fixed point functor. Let $\mathcal P$ be the family of proper subgroups of $G$. There is a classifying $G$-space $E\mathcal P$, and we let $\tilde E \mathcal{P}$ denote the cofiber of the evident based $G$-map $E\mathcal{P}_+ \to S^0$. Any good model is tensored over $G$-spaces, so for any $G$-spectrum $X$, we have the $G$-spectrum $X\wedge \tilde E \mathcal{P}$. Its categorical $G$-fixed point spectrum is the geometric fixed point spectrum of $X$. (This is pointed out briefly in Section XVI.3 of Equivariant Homotopy and Cohomology Theory. I've ignored model theoretic niceties for clarity.)