  Let $k$ be an algebraically closed field, $l\neq\operatorname{char} k$ 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.
