  We study static black hole solutions with locally spherical horizons coupled
to non-Abelian field in $\mathcal{N}=4$ Chern-Simons AdS$_5$ supergravity. They
are governed by three parameters associated to the mass, axial torsion and
amplitude of the internal soliton, and two ones to the gravitational hair. They
describe geometries that can be a global AdS space, naked singularity or a
(non-)extremal black hole. We analyze physical properties of two inequivalent
asymptotically AdS solutions when the spatial section at radial infinity is
either a 3-sphere or a projective 3-space. An important feature of these
3-parametric solutions is that they possess a topological structure including
two $SU(2)$ solitons that wind nontrivially around the black hole horizon, as
characterized by the Pontryagin index. In the extremal black hole limit, the
solitons' strengths match and a soliton-antisoliton system unwinds. That limit
admits both non-BPS and BPS configurations. For the latter, the pure gauge and
non-pure gauge solutions preserve $1/2$ and $1/16$ of the original
supersymmetries, respectively. In a general case, we compute conserved charges
in Hamiltonian formalism, finding many similarities with standard supergravity
black holes.
