  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-year 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 (EUHFORIA) 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 towards 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 maximum
of activity arrives first, while the ICME at minimum of activity is actually
more geo-effective. The helicity sign of the ICME is also a crucial parameter
but at minimum of activity only, since it affects the magnetic profile and the
arrival time of up to 8 hours.
