  We investigate the static and spherically black hole solutions in the
quadratic-order extended vector-tensor theories without suffering from the
Ostrogradsky instabilities, which include the quartic-order
(beyond-)generalized Proca theories as the subclass. We start from the most
general action of the vector-tensor theories constructed with up to the
quadratic-order terms of the first-order covariant derivatives of the vector
field, and derive the Euler-Lagrange equations for the metric and vector field
variables in the static and spherically symmetric backgrounds. We then
substitute the spacetime metric functions of the Schwarzschild,
Schwarzschild-de Sitter/ anti-de Sitter, Reissner-Nordstr\"{o}m-type, and
Reissner-Nordstr\"{o}m-de Sitter/ anti-de Sitter-type solutions and the vector
field with the constant spacetime norm into the Euler-Lagrange equations, and
obtain the conditions for the existence of these black hole solutions. These
solutions are classified into the two cases 1) the solutions with the vanishing
vector field strength; the stealth Schwarzschild and the Schwarzschild de
Sitter/ anti- de Sitter solutions, and 2) those with the nonvanishing vector
field strength; the charged stealth Schwarzschild and the charged Schwarzschild
de Sitter/ anti- de Sitter solutions, in the case that the tuning relation
among the coupling functions is satisfied. In the latter case, if this tuning
relation is violated, the solution becomes the Reissner-Nordstr\"{o}m-type
solution. We show that the conditions for the existence of these solutions are
compatible with the degeneracy conditions for the Class-A theories, and recover
the black hole solutions in the generalized Proca theories as the particular
cases.
