Let $k$ be a field, and $K$ its separable closure. Consider two different $k$-schemes, $X$ and $Y$, which become isomorphic upon extension of scalars to $K$: $X_K \cong Y_K$. Then the etale cohomologies (and indeed, etale homotopy types) of $X_K$ and $Y_K$ will be equivalent, but may differ at $k$. I'd like to know some sample computations in which $H^*_{et}(X_k)$ and $H^*_{et}(Y_k)$ differ, or agree. I'm happy to take any sort of coefficients here, but being an algebraic topologist, I'm lazy, so I would prefer that they're constant sheaves.

To give some focus to the question, let's consider the case $k=\mathbb{R}$, and $K=\mathbb{C}$. Take $X = GL_2(\mathbb{R})$, and let $Y$ be the nonzero quaternions, $\mathbb{H}^{\times}$. These become isomorphic (to $GL_2(\mathbb{C})$) upon extension of scalars to $\mathbb{C}$. Do they have the same etale cohomology over $\mathbb{R}$?

Let's take coefficients such that $2$ is invertible; first note that by a comparison to the analytic topology, $H^*_{et}(GL_2(\mathbb{C}))$ is an exterior algebra on two generators in dimensions 1 and 3, $\Lambda[x_1, x_3]$. There are actions of $\mathbb{Z} / 2 = Gal(\mathbb{C} / \mathbb{R})$ on $H^*_{et}(X_{\mathbb{C}})$ and $H^*_{et}(Y_{\mathbb{C}})$, and the invariants are $H^*_{et}(GL_2(\mathbb{R}))$ and $H^*_{et}(\mathbb{H}^{\times})$, respectively. I presume that the action of $\mathbb{Z} / 2$ on $\Lambda[x_1, x_3]$ must differ for the two examples, but I'm not really sure how to compute these actions.

Lastly, are there any criteria on a scheme under which the etale cohomology over $\mathbb{R}$ bears any resemblance to the singular cohomology of the real points? I have read that this is related to the Sullivan conjecture in homotopy theory, but don't know much more than that slogan.