Abstract The propagation of interplanetary coronal mass ejections (ICMEs) in the heliosphere is influenced by many physical phenomena, related to the internal structure of the ICME and its interaction with the ambient solar wind and magnetic field. As the solar magnetic field is modulated by the 11 yr dynamo cycle, our goal is to perform a theoretical exploratory study to assess the difference of propagation of an ICME in typical minimum and maximum activity backgrounds. We define a median representative CME at 0.1 au, using both observations and numerical simulations, and describe it using a spheromak model. We use the heliospheric propagator EUropean Heliospheric FORecasting Information Asset to inject the same ICME in two different background wind environments. We then study how the environment and the internal CME structure impact the propagation of the ICME toward Earth, by comparison with an unmagnetized CME. At minimum of activity, the structure of the heliosphere around the ecliptic causes the ICME to slow down, creating a delay with the polar parts of the ejecta. This delay is more important if the ICME is faster. At maximum of activity, a southern coronal hole causes a northward deflection. For these cases, we always find that the ICME at the maximum of activity arrives first, while the ICME at the minimum of activity is actually more geoeffective. The sign of the helicity of the ICME is also a crucial parameter, but at the minimum of activity only, since it affects the magnetic profile and the arrival time up to 8 hr.