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This paper presents the results of a $\rm \gamma$-ray likelihood analysisover all the extreme and high synchrotron peak blazars (EHSP & HSP) from the3HSP catalogue. We investigate 2013 multifrequency positions under the eyes ofFermi Large Area Telescope, considering 11 years of observations in the energyrange between 500 MeV to 500 GeV, which results in 1160 $\rm \gamma$-raysignatures detected down to the TS = 9 threshold. The detections include 235additional sources concerning the Fermi Large Area Telescope Fourth SourceCatalog (4FGL), all confirmed via high-energy TS maps, and represent animprovement of ~25% for the number of EHSP & HSP currently described in $\rm\gamma$-rays. We build the $\rm \gamma$-ray spectral energy distribution forall the 1160 2BIGB sources, plot the corresponding $\rm \gamma$-ray logN-logS,and measure their total contribution to the extragalactic gamma-ray background,which reaches up to ~33% at 100 GeV. Also, we show that the $\rm \gamma$-raydetectability improves according to the synchrotron peak flux as represented bythe Figure of Merit (FOM) parameter, and note that the search for TeV peakedblazars may benefit from considering HSP and EHSP as a whole, instead of EHSPsonly. The 2BIGB acronym stands for `Second Brazil-ICRANet Gamma-ray Blazars'catalogue, and all the broadband models and spectral energy distributiondata-points will be available on public data repositories (OpenUniverse,GitHub, and Brazilian Science Data Center-BSDC)

Extreme & High Synchrotron Peak Blazars beyond 4FGL: The 2BIGB $\rm \gamma$-ray catalogue