Relating different Ext groups

Question

If $G$ is a group, $H$ is a normal subgroup, and $A$ and $B$ are $G$-modules, are there any general theorems that relate Ext$_G(A,B)$ to Ext$_H(A,B)$?

Answer 1

If $A$ is free as a $\mathbb{Z}$-module, then $\operatorname{Ext}^\ast_{\mathbb{Z}G}(A,B)\cong H^\ast\left(G,\operatorname{Hom}_{\mathbb{Z}}(A,B)\right)$, and the Lyndon-Hochschild-Serre spectral sequence relates $H^\ast\left(G,\operatorname{Hom}_{\mathbb{Z}}(A,B)\right)$ and $H^\ast\left(H,\operatorname{Hom}_{\mathbb{Z}}(A,B)\right)$.