I am reading now Mike Hill's paper "On the algebras over equivariant little disks" and I have a problem with one (probably) simple equivalence from the proof of the Theorem 2.12.
So let $G$ be a finite group, $H$ its subgroup, $T$ a finite $H$-set and $X$ a $G$-space. The problematic statement is $$ G\times_{H} Map(T,X)\cong G\times_{H}\left( (H\times\Sigma_n /\Gamma_T)\times_{\Sigma_n}X^{\times n}\right). $$ Here $\Gamma_T$ stands for the graph subgroup for $T$, i.e. graph of the homomorphism $H\to \Sigma_n$ defining an $H$-structure on the $T$. And I am not really sure, but I suppose that $Map(T,X)$ just means the set of non-equivariant maps from $T$ to $X$.
Could anybody explain me why this equivalence holds?