Let $k$ be an algebraically closed field, and let $A$ and $B$ be finitely generated $k$-algebras. A map $\varphi:A\to B$ of $k$-algebras induces a map
$$\varphi^*:\mathrm{Ext}_B^*(k,k)\to\mathrm{Ext}_A^*(k,k)$$
What are sufficient conditions on $\varphi$ which make $\mathrm{Ext}_A^*(k,k)$ a finite module over $\mathrm{Ext}_B^*(k,k)$?
This question is motivated by the cohomology of finite groups in which the inclusion of group algebras $kH\hookrightarrow kG$ for a subgroup $H\subset G$ induces a map $H^*(G,k)\to H^*(H,k)$ which gives $H^*(H,k)$ the structure of a finite module over $H^*(G,k)$. I'm trying to isolate the properties of the $k$-algebra map $kH\hookrightarrow kG$ which give a finite map in cohomology so that I can generalize to algebras that are not group algebras of finite groups.