The permutation BG-algebras were first introduced as a novel kind of algebra. In this work, their basic qualities were investigated to better understand how they relate to one another and how they might be combined to construct various sorts of superstructures. We investigated permutation BG-algebras, permutation group-derived, permutation BG-subalgebra, and permutation BG-normal. Furthermore, we explored certain novel concepts in permutation theory for the initial time. We additionally looked into BG-algebra isomorphism theorems, BG-algebra homomorphism, quotient permutation BG-algebras, and equivalence relations.