The present paper discusses the scattering phenomenon of Love-type wave on the functionally graded piezoelectric material (FGPEM) stratum overlying a functionally graded piezomagnetic material (FGPMM) substrate containing abrupt thickening at the uppermost free surface. The expressions for dispersion relation and group velocity (GV) are obtained for four kinds of electric-magnetic boundary conditions. The expressions for mechanical displacement (MD) component, electric potential (EP), and magnetic potential (MP) functions for stratum and substrate are deduced by using the perturbation method and residue theorem. The effect of piezoelectric and piezomagnetic coupling parameters and abrupt thickening parameters on phase velocity (PV), GV, MD component, and EP & MP functions associated with the stratum are graphically represented. A comparative study is carried out to delineate the concealed characteristics of wave on the considered model due to three different shaped abrupt thickening for four kinds of electric-magnetic boundary conditions (i.e. electrically short and magnetically short (ESMS), electrically open and magnetically short (EOMS), electrically open and magnetically open (EOMO), and electrically short and magnetically open (ESMO)).