We　study　the　control　over　spectral　hole　burning　in　four-level　N-type　system　via　spontaneously　generated　coherence　(SGC)　and　Kerr　nonlinearity　in　an　inhomogeneous　Doppler　broadening　medium.　We　consider　a　co-propagating　probe　laser　beam　and　a　counter-propagating　control　laser　beam　at　the　same　time.　Multiple　spectral　holes　are　created　with　the　spontaneously　generated　coherence　in　the　medium.　The　depth　of　the　spectral　holes　gets　deeper　with　increasing　of　the　Kerr　field.　We　study　the　group　index　and　the　transmission　of　probe　laser　pulse　in　SHB　regions　that　lead　us　to　achieve　maximum　transmission　and　much　steeper　group　index　in　the　system　with　increasing　the　strength　of　the　Kerr　field.　©　2020　Elsevier　GmbH