It has been asked [here][1], whether a retract of a finitely presented group is again finitely presented, i.e. if $G$ is a finitely presented group and $H$ is a group which fits into a split exact sequence $1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1$, is then $H$ again finitely presented? The answer is yes and has several solutions (see loc. cit. and in particular [Wall: Finiteness conditions for CW-complexes][2] Lemma 1.3).

A group $G$ is (per def) of type $F_n$ iff there is a model of $K(G,1)$ with a compact $n$-skeleton. $G$ is finitely generated resp. presented iff $G$ is of type $F_1$ resp. $F_2$. So the following question is a generalization of the above: If $H$ is a retract of $G$ and $G$
is of type $F_n$, is then also $H$ of type $F_n$?


  [1]: https://mathoverflow.net/questions/9922/projections-of-finitely-presented-groups
  [2]: http://math.uchicago.edu/~shmuel/tom-readings/wall%20finiteness%201.pdf