  We carry out a systematic study on the motion of test particles in the region
inner to the naked singularity of a quasi--hyperbolically symmetric
$\gamma$-metric. The geodesic equations are written and analyzed in detail. The
obtained results are contrasted with the corresponding results obtained for the
axially symmetric $\gamma$-metric, and the hyperbolically symmetric black hole.
As in this latter case, it is found that test particles experience a repulsive
force within the horizon (naked singularity), which prevents them to reach the
center. However in the present case this behavior is affected by the parameter
$\gamma$ which measures the departure from the hyperbolical symmetry. These
results are obtained for radially moving particles as well as for particles
moving in the $\theta-r$ subspace. Possible relevance of these results in the
explanation of extragalactic jets, is brought out.
