If $0\to A \to B \to C\to 0$ is a short exact sequence of groups and $A$, $C$ are of type $FP_{n}$, then so is $B$. Reference: Proposition 2.2 in <i>Homological finiteness properties of fibre products</i> by Kochloukova and Ferreira Lima ([arXiv link](https://arxiv.org/abs/1611.03759)).