The \'etale symmetric K\"unneth theorem

Let $k$ be an algebraically closed field, $l\mathrm{\ne }\mathrm{char}k$ a prime number, and $X$ a quasi-projective scheme over $k$. We show that the 'etale homotopy type of the $d$th symmetric power of $X$ is $\mathbb Z/l$-homologicallyequivalenttothe$d$th strict symmetric power of the 'etale homotopy type of $X$. We deduce that the $\mathbb{Z}\mathrm{/}l$-local 'etale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.