Section ConsttInertiaBijection.
Variables (gT : finGroupType) (H G : {group gT}) (t : Iirr H).
Hypothesis nsHG : H <| G.
Local Notation theta := 'chi_t.
Local Notation T := 'I_G[theta]%G.
Local Notation "` 'T'" := 'I_(gval G)[theta]
(at level 0, format "` 'T'") : group_scope.
Let calA := irr_constt ('Ind[T] theta).
Let calB := irr_constt ('Ind[G] theta).
Local Notation AtoB := (Ind_Iirr G).
End ConsttIndExtendible.

(* This is Isaacs, Theorem (6.26). *)
Theorem extend_linear_char_from_Sylow G N (lambda : 'CF(N)) :
N <| G -> lambda \is a linear_char -> G \subset 'I[lambda] ->
(forall p, p \in \pi('o(lambda)%CF) ->
exists2 Hp : {group gT},
[/\ N \subset Hp, Hp \subset G & p.-Sylow(G / N) (Hp / N)%g]
& exists u, 'Res 'chi[Hp]_u = lambda) ->
exists u, 'Res[N, G] 'chi_u = lambda.