  The Generalized SU(2) Proca theory is a vector-tensor theory of gravity whose
action is invariant under global transformations of the SU(2) group and
includes second-order derivative self-interactions of the vector field beyond
the massive Yang-Mills theory. We find that the presence of two Lagrangian
pieces consisting of four gauge fields minimally coupled to gravity gives rise
to an exact Reissner-Nordstr\"{o}m black hole solution endowed with two
different non-Abelian effective charges that depend on the specific
combination, $\chi = 2\chi_1 + \chi_2$, of the respective coupling constants.
After studying the spacetime structure of the black hole, some astrophysical
implications of the black hole solutions are investigated. First, joint
analysis of observations of the EHT's first images of Sagittarius A$^{\star}$
of our Galaxy set the first serious constraint on the free parameters of the
theory beyond the theoretical bounds found. Second, we investigate the
accretion properties of spherical steady flows around this class of non-Abelian
Reissner-Nordstr\"{o}m black hole. Specifically, we examine the general
conditions under which transonic flow is allowed. Analytical solution for
critical accretion is found in terms of the coupling constant. In addition, we
explore numerically the effect of changing $\chi$ on the radial velocity and
mass density, and show how the extremal Reissner-Nordstr\"{o}m and the standard
Schwarzschild solutions as limit cases are achieved. Finally, working in the
fully relativistic regime, an analytical expression for the critical mass
accretion rate of a polytropic fluid onto a black hole is derived. As a main
result, we find that the critical accretion rate efficiency can be noticeably
improved compared to the Schwarzschild case for a specific region of the
parameter space where the non-Abelian charge becomes imaginary.
