In algebraic topology, the suspension theorem tells us that for a topological space $X$, we have
$$\tilde{H}^n(X,F)\cong \tilde{H}^{n+k}(S^k\wedge X,F).$$
So I'm wondering if this has an analogue in the category of $\mathbb{A}^1$-homotopy, i.e. do we have
$$H^n\_{ét}(X,F)\cong H^{n+a+b}\_{ét}\left( (\mathbb{A}^1/\{0,1\})^{\wedge a}\wedge (\mathbb{G}\_m)^{\wedge b}\wedge X,F\right)?$$
One thing that would make me think that this is true is the comparison theorem for étale cohomology and singular cohomology. Indeed, we have
$$\left(\mathbb{A}^1/\{0,1\}\right)(\mathbb{C})^{an}\cong S^1\cong \mathbb{G}\_m(\mathbb{C})^{an}$$
and thus we have
$$H^{n+a+b}\_{ét}\left( (\mathbb{A}^1/\{0,1\})^{\wedge a}\wedge (\mathbb{G}\_m)^{\wedge b}\wedge X,F\right)\cong H^{n+a+b}((S^1)^{\wedge(a+b)}\wedge X(\mathbb{C}),F)\cong H^{n}(X(\mathbb{C})^{an},F)\cong H^n\_{ét}(X,F)$$
if we assume that $F$ is finite so that we can apply the comparision theorem. Is this true? If this argument does not work, is there another proof?



Suspension Theorem in $\mathbb{A}^1$-homotopy
This question has already been satisfactorily answered in the comments. To avoid it lingering around as "unanswered," I am providing a CW answer with a reference for the observations made by Denis-Charles Cisinski.


A complete treatment of etale cohomology operations is provided in [Operations in étale and motivic cohomology](https://arxiv.org/pdf/1301.0872.pdf), where, as the authors write in the abstract "We classify all étale cohomology operations." This includes lots of pointers to the early literature, by Epstein, Morel, Voevodsky, and others. For results of the form stated by Cisinski, see, e.g., section 7. For a result along the lines of the "suspension theorem" referenced in the OP, see page 5.


