An extension of algebras is a homomorphism that preserves identities between algebras. In this talk, we shall compare the relative homological dimensions of two extensions under certain conditions, including relative global dimensions and relative weak dimensions of extensions. Then we provide a sufficient condition for two extensions to have the same relative global dimensions, and our result can be applied to compare the relative global dimensions of relative Hochschild-Nagata extensions. Furthermore, for any natural number n, we present a general method for constructing extensions of Artin algebras with a relative global dimension equal to n. Additionally, we will briefly introduce the theory of relative Gorensteion homological algebra on extensions.