Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the field with two elements.
For $V$ an $\mathbb{F}_2$-module, hence an $\mathbb{F}_2$-vector space, and for $n \in \mathbb{N}$, write
for the homotopy quotient of the $n$-fold tensor product of $V$ with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a free resolution $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:
This is called the $n$th extended power of $V$.
Jacob Lurie, 18.917 Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007. (MIT OpenCourseWare: Massachusetts Institute of Technology), Lecture notes
Lecture 2 Steenrod operations (pdf)
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