  AIMS. We are studying an unbiased sample of 42 nearby (z < 0.2) BL Lacertae
objects with a multi-wavelength approach. The results of VLBI observations were
presented in the first paper of this series. In this paper, we study the
$\gamma$-ray properties of the sample.
  METHODS. We analyse data collected by the Fermi LAT during its first 8.5
years of operation in the energy range 0.1-300 GeV.
  RESULTS. We reveal 23 sources with a test statistic greater than 25
(corresponding to $\sim$ 4.6-$\sigma$) out of 42, with 3 sources not detected
in the 3LAC catalogue, and fluxes between $3.5\times 10^{-10}$ and
$7.4\times10^{-8}$ ph cm$^{-2}$ s$^{-1}$. The majority of the sources have hard
spectra ($\Gamma \leq 2$), with only four having values in the range 2.1-2.4.
The three newly detected sources have fluxes in the range between
$0.54\times10^{-9}$ and $1.35\times10^{-9}$ ph cm$^{-2}$ s$^{-1}$, and photon
index 1.7-1.9. Among the 23 LAT-detected sources, 19 are included in the 3FHL,
with a spectrum that connects relatively smoothly from 0.1 GeV to 2 TeV.
LAT-detected BL Lacs are more luminous on parsec scales with respect to
non-LAT-detected sources and have larger core dominance according to the
unified models.
  CONCLUSIONS. The LAT-detected BL Lacs seem to be composed of a bulk of
"classical" sources dominated by Doppler boosting and characterised by compact
and bright radio emission as well as hard $\gamma$-ray spectra. Moreover, we
have identified a possible population of low-luminosity BL Lacs not detected by
LAT, lacking a VLBI core, and with a small Doppler factor. Furthermore, three
LAT-detected sources show non-classical properties for $\gamma$-ray emitting BL
Lacs (no evidence of relativistic jet, low Doppler factor in radio images,
relatively low core dominance) and three other sources, while showing radio
emission on parsec scales, are not detected in $\gamma$ rays so far.
