Suppose $C$ is a symmetric monoidal category with monoidal product $\wedge$, $X$ is a $G$-object for some finite group $G$ (say), and $T$ is a finite $G$-set of size $n$. The $n$-fold monoidal power $X \wedge \dots \wedge X$ carries a $S_n \wr G$-action ($S_n$ the symmetric group on $n$ letters). Since $T$ is a $G$-set we have an essentially canonical map $G \to S_n$ (up to choosing a bijection of sets between $T$ and $\{1, \dots, n\}$), and hence also the diagonal $G \to S_n \wr G.$ This way we can give the $n$-fold smash power an "exotic" $G$-structure; denote this by $X^{\wedge T}.$
I am wondering if this sort of structure has been studied, and if anything interesting can be said, perhaps in particular cases. If it has been studied, under what name is this the case?
For example, if $C$ is the category of $R$-modules and $\wedge = \oplus,$ then for a representation $V$ we have $V^{\oplus T} = Perm(T) \otimes V$ so we get nothing new. On the other hand if we put $\wedge = \otimes,$ this construction seems to be non-trivial in general. Does it have a more familiar name? Note that $V^{\oplus T}$ can be generalised to "exponentiating by representations" via $V^{\oplus W} := W \otimes V.$ Can we make sense of "$V^{\otimes W}$"? Writing down some explicit expressions using a basis would suggest yes, but this is getting messy. Can we see on the level of general symmetric monoidal categories why/when such an extension (from exponentiating by $G$-sets to exponentiating by representations) should be possible?
I think this is studied in equivariant homotopy theory. I've been told there is some interest in defining something like $G$-spaces $X$ (or maybe $G$-spectra) which are equipped with not only multiplications $X^n \to X$ for all natural numbers $n$ but for all finite $G$-sets. I'm not sure what the right keywords are though (I vaguely remember "genuinely commutative"?); you should ask in the MO homotopy theory chat.