Canonical Fitting_igFun := [igFun by Fitting_sub & morphim_Fitting].
Canonical Fitting_gFun := [gFun by morphim_Fitting].
Canonical Fitting_pgFun := [pgFun by morphim_Fitting].
Section IsoFitting.
Variables (gT rT : finGroupType) (G D : {group gT}) (f : {morphism D >-> rT}).

Lemma charsimple_dprod G : charsimple G ->
exists H : {group gT}, [/\ H \subset G, simple H
& exists2 I : {set {perm gT}}, I \subset Aut G
& \big[dprod/1]_(f in I) f @: H = G].