A covariant Stinespring type theorem for $\tau$-maps

Let $\tau$ be a linear map from a unital $C^{\ast }$-algebra $\text{CMcal}A$ to a von Neumann algebra $\text{mathematical}B$ and let $\text{CMcal}C$ be a unital $C^{\ast }$-algebra. A map $T$ from a Hilbert $\text{CMcal}A$-module $E$ to a von Neumann $\CMcal C$-$\CMcal B$module$F$iscalleda$\tau$-mapif$$\langle T(x),T(y)\rangle=\tau(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ 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 $(u^{\mathrm{\prime }},u)$-covariant with respect to a dynamical system $(G,\eta ,E)$ and the set of all $(u^{\mathrm{\prime }},u)$-covariant $\tilde{\tau }$-maps from the crossed product $E{\times }_{\eta }G$ to $F$, where $\tau$ and $\tilde{\tau }$ are completely positive.