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Let $\tau$ be a linear map from a unital $C^*$-algebra $\CMcal A$ to a vonNeumann 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 $\CMcalC$-$\CMcal B$ module $F$ is called a $\tau$-map if $$\langleT(x),T(y)\rangle=\tau(\langle x, y\rangle)~\mbox{for all}~x,y\in E.$$ AStinespring type theorem for $\tau$-maps and its covariant version are obtainedwhen $\tau$ is completely positive. We show that there is a bijectivecorrespondence 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 setof all $(u',u)$-covariant $\widetilde{\tau}$-maps from the crossed product$E\times_{\eta} G$ to $F$, where $\tau$ and $\widetilde{\tau}$ are completelypositive.

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