A covariant Stinespring type theorem for $τ$-maps

Let $\tau$ be a linear map from a unital $C^*$-algebra $\CMcal A$ to a von Neumann algebra $\mathematical B$ and let $\CMcal C$ be a unital $C^*$-algebra. A map $T$ from a Hilbert $\CMcal A$-module $E$ to a von Neumann $\CMcal C$-$\CMcal B$ module $F$ is called a $\tau$-map if $$\langle T(x),T(y)\rangle=\tau(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ A Stinespring type theorem for $\tau$-maps and its covariant version are obtained when $\tau$ is completely positive. We show that there is a bijective correspondence between the set of all $\tau$-maps from $E$ to $F$ which are $(u',u)$-covariant with respect to a dynamical system $(G,\eta,E)$ and the set of all $(u',u)$-covariant $\widetilde{\tau}$-maps from the crossed product $E\times_{\eta} G$ to $F$, where $\tau$ and $\widetilde{\tau}$ are completely positive.