It is known that quotients of finitely generated groups are finitely generated and that the quotient of a finitely presented group is finitely presented iff the normal subgroup is the normal closure of a finite subset of the group. Hence: $$G~\text{is }F_1\Rightarrow G/N~\text{is }F_1 \forall~N\trianglelefteq G$$ $$G~\text{is }F_2\Leftrightarrow G/N~\text{is }F_2 \forall~N\trianglelefteq G\text{ which are normaly finitely generated}$$ My question is about the higher dimensional finiteness properties. Are there any similar statements about the quotient of an $F_n$ group being $F_n$?