Let $R$ be an Artin algebra, $\mathrm{m}\mathrm{o}\mathrm{d}R$ be the category of finitely generated right $R$-modules, and let $F$ be a sub-bifunctor of ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^1(-,-)$. In this paper, we introduce the notion of the $F$-extension dimension for $\mathrm{m}\mathrm{o}\mathrm{d}R$. We further establish the relationships for the finiteness among the $F$-extension dimension, $F$-finitistic dimension and $F$-Igusa-Todorov dimension.