Let X be a space with $Z/2$ action. There is a map from $K(X)$ to the equivariant K-group $K_{Z_{2}}(X)$, which is called "the induction map". (It is a standard operation in equivariant stable homotopy theory which maps $\pi_{H}(X)$ to $\pi_{G}(X)$ for $H$ a finite index subgroup in $G$.) It is important when we consider the Gysin sequence for equivariant stable homotopy theory.

My question is: does this map just map an ordinary bundle $E\in K(X)$ to the $Z_{2}$-bundle $E\oplus\tau^{*}(E)\in K_{Z_{2}}(X)$? Here $\tau$ is the involution defined by the $Z_{2}$ action. I think this is the case but I can't find the reference.