Quotient bifinite extensions and the finitistic dimension conjecture

We prove that if $B\subseteq A$ is an extension of finite dimensional algebras such that the projective dimension of $A\mathrm{/}B$ as a $B$-bimodule is finite, if $A$ has finite finitistic dimension, then so does $B$. We exhibit examples demonstrating that the algebra $B$ appearing in such an extension can be more complicated than $A$.