  Horndeski's vector-tensor (HVT) gravity is described by a Lagrangian in which
the field strength $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$
of a vector field $A_{\mu}$ interacts with a double dual Riemann tensor $L^{\mu
\nu \alpha \beta}$ in the form $\beta L^{\mu \nu \alpha \beta} F_{\mu \nu}
F_{\alpha \beta}$, where $\beta$ is a constant. In Einstein-Maxwell-HVT theory,
there are static and spherically symmetric black hole (BH) solutions with
electric or magnetic charges, whose metric components are modified from those
in the Reissner-Nordstr\"om geometry. The electric-magnetic duality of
solutions is broken even at the background level by the nonvanishing coupling
constant $\beta$. We compute a second-order action of BH perturbations
containing both the odd- and even-parity modes and show that there are four
dynamical perturbations arising from the gravitational and vector-field
sectors. We derive all the linear stability conditions associated with the
absence of ghosts and radial/angular Laplacian instabilities for both the
electric and magnetic BHs. These conditions exhibit the difference between the
electrically and magnetically charged cases by reflecting the breaking of
electric-magnetic duality at the level of perturbations. In particular, the
four angular propagation speeds in the large-multipole limit are different from
each other for both the electric and magnetic BHs. This suggests the breaking
of eikonal correspondence between the peak position of at least one of the
potentials of dynamical perturbations and the radius of photon sphere. For the
electrically and magnetically charged cases, we elucidate parameter spaces of
the HVT coupling and the BH charge in which the BHs without naked singularities
are linearly stable.
