  We use collisionless $N$-body simulations to determine how the growth of a
supermassive black hole (SMBH) influences the nuclear kinematics in both barred
and unbarred galaxies. In the presence of a bar, the increase in the velocity
dispersion $\sigma$ (within the effective radius) due to the growth of an SMBH
is on average $\lesssim 10%$, whereas the increase is only $\lesssim 4%$ in an
unbarred galaxy. In a barred galaxy, the increase results from a combination of
three separate factors (a) orientation and inclination effects; (b) angular
momentum transport by the bar that results in an increase in the central mass
density; (c) an increase in the vertical and radial velocity anisotropy of
stars in the vicinity of the SMBH. In contrast the growth of the SMBH in an
unbarred galaxy causes the velocity distribution in the inner part of the
nucleus to become less radially anisotropic. The increase in $\sigma$ following
the growth of the SMBH is insensitive to a variation of a factor of 10 in the
final mass of the SMBH, showing that it is the growth process rather than the
actual SMBH mass that alters bar evolution in a way that increases $\sigma$. We
argue that using an axisymmetric stellar dynamical modeling code to measure
SMBH masses in barred galaxies could result in a slight overestimate of the
derived $M_{BH}$, especially if a constant M/L ratio is assumed. We conclude
that the growth of a black hole in the presence of a bar could result in an
increase in $\sigma$ which is roughly of $4-8%$ larger than the increase that
occurs in an axisymmetric system. While the increase in $\sigma$ due to SMBH
growth in a barred galaxy might partially account for the claimed offset of
barred galaxies and pseudo bulges from the $M_{BH}-\sigma$ relation obtained
for elliptical galaxies and classical bulges in unbarred galaxies, it is
inadequate to account for all of the offset.
