Supersoluble immersion of a normal subgroup K of a finite group G shall be defined by the following property:
If σ is a homomorphism of G, and if the minimal normal subgroup J of Gσ is part of Kσ then J is cyclic (of order a prime).
Our principal aim in the present investigation is the proof of the equivalence of the following three properties of the normal subgroup K of the finite group G:
(i) K is supersolubly immersed in G.
(ii) K/ϕK is supersolubly immersed in G/ϕK.
(iii) If θ is the group of automorphisms induced in the p-subgroup U of K by elements in the normalizer of U in G, then θ' θp-1 is a p-subgroup of θ.
Though most of our discussion is concerned with the proof of this theorem, some of our concepts and results are of independent interest.