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A study of the $p d \to p d \eta$ reaction in the energy range where therecent data from Uppsala are available, is done in the two-step model of $\eta$production including the final state interaction. The $\eta -d$ final stateinteraction is incorporated through the solution of the Lippmann Schwingerequation using an elastic scattering matrix element, $T_{\eta d \to \eta d}$,which is required to be half off-shell. It is written in a factorized form,with an off-shell form factor multiplying an on-shell part given by aneffective range expansion up to the fourth power in momentum. The parameters ofthis expansion have been taken from an existing recent relativistic Faddeevequation solution for the $\eta NN$ system corresponding to different $\eta-N$scattering amplitudes. Calculations have also been done using few bodyequations within a finite rank approximation (FRA) to generate $T_{\eta d \to\eta d}$. The $p-d$ final state interaction is included in the spirit of theWatson-Migdal prescription by multiplying the matrix element by the inverse ofthe Jost function. The $\eta-d$ interaction is found to be dominant in theregion of small invariant $\eta -d$ mass, $M_{\eta d}$. The $p-d$ interactionenhances the cross section in the whole region of $M_{\eta d}$, but is largerfor large $M_{\eta d}$. We find nearly isotropic angular distributions of theproton and the deuteron in the final state. All the above observations are inagreement with data. The production mechanism for the entire range of theexisting data on the $p d \to p d \eta$ reaction seems to be dominated by thetwo-step model of $\eta$ production.