This is to address Yonatan's question in the comments. Let $G$ be a finite group. To every (genuine) $G$-spectrum $E$, you can associate its (genuine) fixed point spectrum $E^{G}$. This earns its name by virtue of the following compatibility between stable and unstable homotopy theory:
$\Omega^{\infty}( E^G ) = (\Omega^{\infty} E)^{G}$.

Now you could ask, could we have some categorically dual gadget
$E \mapsto E_{G}$, which satisfied the dual condition
$\Sigma^{\infty}_{+} (X/G) = (\Sigma^{\infty}_{+} X)_{G}$ for $X$ a $G$-space? The answer is no, because the construction on the right hand side
$X \mapsto \Sigma^{\infty}_{+} (X/G)$ is functorial with respect to maps of $G$-spaces, but not with respect to stable maps of $G$-spaces.

For example, any correspondence of $G$-spaces $X \leftarrow M \rightarrow Y$ where $M$ is a finite covering space of $X$ determines a map
of $G$-spectra from $\Sigma^{\infty}_{+}(X)$ to $\Sigma^{\infty}_{+}(Y)$.
But there's no functorial procedure to extract from $M$ a map of spectra
$\Sigma^{\infty}_{+} (X/G) \rightarrow \Sigma^{\infty}_{+}(Y/G)$.
You might say: why not use the correspondence of nonequivariant spaces given by $X/G \leftarrow M/G \rightarrow Y/G$? This does not respect composition of correspondences (observe what happens when you compose the correspondence $\ast \leftarrow G \rightarrow \ast$ with itself).