Let $f: X\to Y$ be a morphism of schemes. If it is needed, we can assume that $X,Y$ are quasi-compact and quasi-separated, and $f$ is affine. Consider the two small etale sites $X_{et}$ and $Y_{et}$. Let $\mathcal{F}$ be an abelian sheaf over $Y_{et}$. My question is about the canonical homomorphism: $$\varphi:\ H^q(Y_{et};Y,\mathcal{F})\to H^q(X_{et};X,f^*\mathcal{F})$$ which is induced by $\Gamma(Y,\mathcal{F})\to \Gamma(Y,f_*f^*\mathcal{F})=\Gamma(X,f^*\mathcal{F})$.
Is $\varphi$ an isomorphism?
I tried to show that $\Gamma(Y,-)=\Gamma(X,-)\circ f^*$, but I failed. I can only show that $\Gamma(Y,-)=\Gamma(X,-)\circ f^*$ when $f$ is etale, which is a trivial result. And I also know that this will be true if we consider the cohomology of quasi-coherent sheafs and assume that $f:X\to Y$ is an affine morphism since $f_*$ is exact.
Is this true for affine morphism between quasi-compact, quasi-separated schemes? And general cases? Can you help me?