The etale symmetric Kunneth theorem

Let $k$ be an algebraically closed field, $l eqoperatorname{char} k$ 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 $mathbb Z/l$-homologically equivalent to the $d$th strict symmetric power of the etale homotopy type of $X$. We deduce that the $mathbb Z/l$-local etale homotopy type of a motivic Eilenberg-Mac Lane space is an ordinary Eilenberg-Mac Lane space.