  We introduce and study a fractional version of the Skellam process of order
$k$ by time-changing it with an independent inverse stable subordinator. We
call it the fractional Skellam process of order $k$ (FSPoK). An integral
representation for its one-dimensional distributions and their governing system
of fractional differential equations are obtained. We derive the probability
generating function, mean, variance and covariance of the FSPoK which are
utilized to establish its long-range dependence property. Later, we considered
two time-changed versions of the FSPoK. These are obtained by time-changing the
FSPoK by an independent L\'evy subordinator and its inverse. Some
distributional properties and particular cases are discussed for these
time-changed processes.
