Let $U$ be a representation of $S\_m$ and $V$ a representation of $S\_n$. Then the representation $\operatorname{Ind}\_{S\_m\wr S\_n}^{S\_{mn}}(U^{\otimes{n}}\otimes V)$ has a nice interpretation in terms of symmetric functions. If $\operatorname{ch}(U)$ and $\operatorname{ch}(V)$ are the Frobenius characteristics of $U$ and $V$ (symmetric functions of degree $m$ and $n$), then
$$\operatorname{ch}\operatorname{Ind}\_{S\_m\wr S\_n}^{S\_{mn}}(U^{\otimes{n}}\otimes V)= \operatorname{ch}(V)[\operatorname{ch}(U)],$$
where the square brackets denote plethysm. I am interested in a graded version of this result.


More specifically, let $X$ be a space with an action of $S\_m$ and $Y$ a space with an action of $S\_n$. I am interested in the representation
$$\operatorname{Ind}\_{S\_m\wr S\_n}^{S\_{mn}}\left(H^\*(X^n)\otimes H^\*(Y)\right)$$ of $S\_{mn}$. The naive statement would be that
$$\operatorname{ch}\operatorname{Ind}\_{S\_m\wr S\_n}^{S\_{mn}}\left(H^\*(X^n)\otimes H^\*(Y)\right) = \operatorname{ch}(H^\*(Y))[\operatorname{ch}(H^\*(X))],$$
where $\operatorname{ch}(H^\*(X))$ and $\operatorname{ch}(H^\*(Y))$ are symmetric functions of degree $m$ and $n$ with coefficients in the polynomial ring $\mathbb{Z}[t]$. 


I'm pretty sure that this statement is correct when the cohomology of $X$ is concentrated in even degree, but otherwise it is wrong. The issue is that, at the level of $S\_n$-representations, the Kunneth isomorphism $H^\*(X^n)\cong H^\*(X)^{\otimes n}$ needs to be interpreted in terms of super-vector spaces: the action of the simple transposition $(i,i+1)$ picks up a sign when the two cohomology classes being swapped have odd degree. For example, if $X=S^1$ and $n=2$, then $H^2(S^1\times S^1)\cong H^1(S^1)\otimes H^1(S^1)$ is the sign representation of $S\_2$ rather than the trivial representation.


So my question is: Is there a "super version" of plethysm for symmetric functions with coefficients in $\mathbb{Z}[t]$ for which the last displayed equation is correct? 


I know that the answer is tautologically "yes"--one can just translate from symmetric functions to representations, do the induction, and translate that, and take this as the definition of super-plethysm. But I'm looking for something explicit enough to allow me to do calculations in SAGE.



Super-plethysm?
This amounts to study composition of "linear species" in the category of complexes.


The correct way to handle these computations using plethysm is
to introduce an auxiliary variable $t$ and to weight the cohomology $H^i$ with the weight $(-t)^i$. The plethysm must act on $t$ by $p\_n(t)=t^n$.


I do not know a written reference.


