  We study a Kerr-like black hole and naked singularity in perfect fluid dark
matter (PFDM). The critical value of spin parameter $a_c$ is presented to
differentiate the black hole from naked singularity. It is seen that for any
fixed value of dark matter parameter $\alpha$ the rotating object is black hole
if $a\leq a_c$ and naked singularity if $a>a_c$. Also for $-2\leq\alpha<2/3$
the size of the black hole horizons decrease whereas for $2/3<\alpha$ it
increases. We also study spin precession frequency of a test gyroscope attached
to stationary observer to differentiate a black hole from naked singularity in
perfect fluid dark matter. For the black hole, spin precession frequency blows
up as the observer reaches the central object while for naked singularity it
remains finite except at the ring singularity. Moreover, we study
Lense-Thirring precession for a Kerr-like black hole and geodetic precession
for Schwarzschild black hole in perfect fluid dark matter. To this end, we have
calculated the Kepler frequency (KF), the vertical epicyclic frequency (VEF),
and the nodal plane precession frequency (NPPF). Our results show that, the
PFDM parameter $\alpha$ significantly affects those frequencies. This
difference can be used by astrophysical observations in the near future to shed
some light on the nature of dark matter.
