On the calculation of trace norms of maps generated by projective unitary representations of finite groups

Let $G\ni g\to U_g\in {\mathcal U}(H)$ be an irreducible projective unitary representation of a finite group $G$ in a Hilbert space $H$. Denote $\mathfrak {S}(H)$ the convex set of positive operators with unit trace in $H$ (quantum states). We consider maps $\Phi :\mathfrak {S}(H)\to \mathfrak {S}(H)$ having the forms
$$ \Phi (\rho )=\sum \limits _{g\in G}\pi _gU_g\rho U_g^*,\ \rho \in \mathfrak {S}(H), $$
where $(\pi _g)$ are probability distributions on the group $G$. We study the question of finding certain class of functions from $\Phi$ including trace norms of the form
$$ ||\Phi ||_{1\to p}=[\sup \limits _{\rho \in \mathfrak {S}(H)}Tr(\Phi (\rho )^p)]^{\frac {1}{p}},\ p>1, $$
as for $\Phi$ and its tensor powers $\Phi ^{\otimes n}$ as well.