  Interacting dark matter-dark energy (IDMDE) models can be taken to account as
one of the present challenges that may affect the cosmic structures. In this
work, we study the integrated Sachs-Wolfe (ISW) effect in IDMDE models. To this
end, we initially introduce a theoretical framework for IDMDE models. Moreover,
we briefly discuss the stability conditions of IDMDE models and by specifying a
simple functional form for the energy density transfer rate, we calculate the
perturbation equations. In the following, we calculate the amplitude of the
matter power spectrum for the IDMDE model and compare it with the corresponding
result obtained from the $\Lambda$CDM model. Furthermore, we calculate the
amplitude of the ISW auto-power spectrum as a function of multipole order l for
the IDMDE model. The results indicate that the amplitude of the ISW auto-power
spectrum in the IDMDE model for different phantom dark energy equations of
state behaves similar to the one for the $\Lambda$CDM model, whereas, for the
quintessence dark energy equations of state, the amplitude of the ISW-auto
power spectrum for the IDMDE model should be higher than the one for the
$\Lambda$CDM model. Also, it turns out that the corresponding results by
different values of the coupling parameter demonstrate that $\xi$ is inversely
proportional to the amplitude of the ISW-auto power spectrum in the IDMDE
model. Finally, by employing four different surveys, we calculate the amplitude
of the ISW-cross power spectrum as a function of multipole order $l$ for the
IDMDE model. The results exhibit that the amplitude of the ISW-cross power
spectrum for the IDMDE model for all values of $\omega_{\rm x}$ is higher than
the one obtained for the $\Lambda$CDM model. Also, it turns out that the
amplitude of the ISW-cross power spectrum in the IDMDE model changes inversely
with the value of coupling parameter $\xi$.
