This paper presents the results of a {gamma}-ray likelihood analysis over all the extreme and high synchrotron peak blazars (EHSP and HSP) from the 3HSP catalogue. We investigate 2013 multifrequency positions under the eyes of Fermi Large Area Telescope, considering 11yr of observations in the energy range between 500MeV and 500GeV, which results in 1160 {gamma}-ray signatures detected down to the TS=9 threshold. The detections include 235 additional sources concerning the Fermi Large Area Telescope Fourth Source Catalog (4FGL), all confirmed via high-energy TS (Test Statistic) maps, and represent an improvement of ~25% for the number of EHSP and HSP currently described in {gamma}-rays. We build the {gamma}-ray spectral energy distribution (SED) for all the 1160 2BIGB sources, plot the corresponding {gamma}-ray logN-logS, and measure their total contribution to the extragalactic gamma-ray background, which reaches up to ~33% at 100GeV. Also, we show that the {gamma}-ray detectability improves according to the synchrotron peak flux as represented by the figure of merit parameter, and note that the search for TeV peaked blazars may benefit from considering HSP and EHSP as a whole, instead of EHSPs only. The 2BIGB acronym stands for 'Second Brazil-ICRANet Gamma-ray Blazars' catalogue.