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 $3.5times 10^{-10}$ and $7.4times10^{-8}$ 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 $0.54times10^{-9}$ and $1.35times10^{-9}$ 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.
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